parameterized complexity of distances to sparse graph classes · this dissertation o ers...
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Technical ReportNumber 903
Computer Laboratory
UCAM-CL-TR-903ISSN 1476-2986
Parameterized complexity ofdistances to sparse graph classes
Jannis Bulian
February 2017
15 JJ Thomson AvenueCambridge CB3 0FDUnited Kingdomphone +44 1223 763500
http://www.cl.cam.ac.uk/
c© 2017 Jannis Bulian
This technical report is based on a dissertation submittedFebruary 2016 by the author for the degree of Doctor ofPhilosophy to the University of Cambridge, Clare College.
Technical reports published by the University of CambridgeComputer Laboratory are freely available via the Internet:
http://www.cl.cam.ac.uk/techreports/
ISSN 1476-2986
Abstract
This dissertation offers contributions to the area of parameterized complexity theory,which studies the complexity of computational problems in a multivariate framework.We demonstrate that for graph problems, many popular parameters can be understoodas distances that measure how far a graph is from belonging to a class of sparse graphs.We introduce the term distance parameter for such parameters and demonstrate theirvalue in several ways.
The parameter tree-depth is uncovered as a distance parameter, and we establishseveral fixed-parameter tractability and hardness results for problems parameterized bytree-depth.
We introduce a new distance parameter elimination distance to a class C. The param-eter measures distance in a way that naturally generalises the notion of vertex deletion byallowing deletions from every component in each deletion step. We show that tree-depthis a special case of this new parameter, introduce several characterisations of eliminationdistance, and discuss applications. In particular we demonstrate that Graph Isomor-phism is FPT parameterized by elimination distance to bounded degree, extending knownresults. We also show that checking whether a given graph has elimination distance kto a minor-closed class C is FPT, and, moreover, if we are given a finite set of minorscharacterising C, that there is an explicit FPT algorithm for this.
We establish an algorithmic meta-theorem for several distance parameters, by showingthat all graph problems that are both slicewise nowhere dense and slicewise first-orderdefinable are FPT.
Lastly, several fixed-parameter tractability results are established for two graph prob-lems that have natural distance parameterizations.
Acknowledgements
First and foremost I thank my supervisor, Anuj Dawar, whose advice has been invaluableto me. I could not have asked for a more patient, accessible and supportive mentor. Ialso thank Timothy Griffin and Andrew Pitts for their helpful feedback on my first-yearreport.
This work would not have been possible without the financial support from the Qual-comm Research Scholarship and the EPSRC.
I thank Pengming Wang and Nik Sultana for their comments on a draft of this disser-tation.
I also thank my parents for their continuous support throughout my years of study(and my life in general). Lastly, I am particularly grateful to Doerte Letzmann for herinvaluable support over the last decade.
Contents
1 Introduction 111.1 Related areas of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2 Contribution details . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 General background 212.1 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 Second-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Parameterized complexity theory . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Fixed-parameter tractability . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Kernelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.3 The W[i] hierarchy and the class XP . . . . . . . . . . . . . . . . . 252.2.4 Model theoretic definitions . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.1 Graph isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Graph minor theory . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3 Nowhere dense & bounded expansion graph classes . . . . . . . . . 29
2.4 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Distance parameters for graph problems 333.1 Deletion distance to a class C . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Vertex cover number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Edit distance to a class C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Path-width & tree-width . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Clique-width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Max leaf number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.7 Degeneracy and the k-core of a graph . . . . . . . . . . . . . . . . . . . . . 39
4 Tree-depth: Exploring the space between vertex cover number and tree-width 414.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Alternative characterisations of tree-depth . . . . . . . . . . . . . . . . . . 424.3 Relation to vertex cover number and tree-width . . . . . . . . . . . . . . . 444.4 Generalised tree-depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5.1 List Colouring and Precolouring Extension . . . . . . . . 46
4.5.2 Approximation and Parameterized complexity: Boxicity . . . . . 484.5.2.1 The algorithm for Boxicity . . . . . . . . . . . . . . . . 50
4.6 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Elimination distance 555.1 Elimination distance to a class C . . . . . . . . . . . . . . . . . . . . . . . 575.2 Alternative characterisations of elimination distance . . . . . . . . . . . . . 58
5.2.1 Elimination order to C . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.2 Separation into two parts . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.2.1 Characterisation via games . . . . . . . . . . . . . . . . . 625.2.3 Apex graphs and closures under disjoint unions . . . . . . . . . . . 62
5.3 Elimination distance & classes of bounded expansion . . . . . . . . . . . . 635.4 Elimination distance to the class of edgeless graphs . . . . . . . . . . . . . 64
5.4.1 Elimination distance & generalised tree-depth . . . . . . . . . . . . 655.5 Elimination distance to bounded degree classes . . . . . . . . . . . . . . . . 65
6 Graph Isomorphism 696.1 Isomorphism on bounded-degree graphs . . . . . . . . . . . . . . . . . . . . 736.2 Deletion distance to bounded degree . . . . . . . . . . . . . . . . . . . . . 746.3 Elimination distance to bounded degree . . . . . . . . . . . . . . . . . . . . 76
6.3.1 Canonical separation into high and low degree parts . . . . . . . . . 776.3.2 Graph canonisation algorithm . . . . . . . . . . . . . . . . . . . . . 81
7 Graph classes characterised by excluded minors 877.1 Elimination distance to excluded minors . . . . . . . . . . . . . . . . . . . 887.2 Computing the excluded minors of Capex and C . . . . . . . . . . . . . . . . 897.3 Computing the excluded minors of Ck . . . . . . . . . . . . . . . . . . . . . 91
8 Slicewise first-order and slicewise nowhere-dense classes 938.1 Evaluating formulas on nowhere dense classes . . . . . . . . . . . . . . . . 948.2 Deciding first-order definable nowhere dense problems . . . . . . . . . . . . 978.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.3.1 Deletion distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.3.2 Edit distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.3.3 Tree-depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
9 Distance parameters to sparse graph classes: two case studies 1059.1 Clique Dominating Set . . . . . . . . . . . . . . . . . . . . . . . . . . 1069.2 The Anchored k-core problem . . . . . . . . . . . . . . . . . . . . . . . 108
9.2.1 Parameters: budget and tree-width . . . . . . . . . . . . . . . . . . 1109.2.2 Parameters: degree, size of the core, and clique-width . . . . . . . . 1119.2.3 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10 Further remarks and future research directions 11310.1 Future research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.1.1 Elimination distance . . . . . . . . . . . . . . . . . . . . . . . . . . 11410.1.1.1 Graph Isomorphism . . . . . . . . . . . . . . . . . . . . 11410.1.1.2 Elimination distance to bounded degree . . . . . . . . . . 114
10.1.1.3 Elimination distance to edgeless graphs . . . . . . . . . . . 11510.1.2 Algorithmic meta theorems . . . . . . . . . . . . . . . . . . . . . . 115
10.1.2.1 Graph classes including dense graphs . . . . . . . . . . . . 11510.1.2.2 Order-invariant first-order logic . . . . . . . . . . . . . . . 116
10.1.3 Descriptive complexity . . . . . . . . . . . . . . . . . . . . . . . . . 11610.1.4 Subquadratic FPT algorithms . . . . . . . . . . . . . . . . . . . . . 117
Bibliography 119
Index 131
CHAPTER 1
Introduction
The question which computational tasks can be efficiently solved in a systematic way hasbeen studied for a long time. The first algorithms date back as far as ancient Greece– famous examples include the Sieve of Eratosthenes for identifying prime numbers andEuclid’s algorithm for finding the greatest common divisor of two numbers. Today, withcomputers being ubiquitous, algorithms are more important than ever. One of the centralquestions is: what computational tasks (usually called problems) can we expect to solvealgorithmically. Or, which problems can we practically solve on a computer?
Alan Turing showed that there are limits to what an algorithm can do. Turing [152]famously formalised computation by introducing Turing machines, and proved that thereare problems that can be solved by no algorithm: He showed that the halting problemfor Turing machines is undecidable – that there is no algorithm that, taking anotheralgorithm as an input, can decide whether that input algorithm will ever halt. He usedthis to show that Hilbert’s ‘Entscheidungsproblem’ is undecidable.
Turing pointed out a fundamental limit of computation, and other problems have beenshown to be uncomputable. However, even if we know that a computational problem canin principle be solved by an algorithm, it might still not be feasible. Solving the problemmight require more computational resources (i.e. space or time) than available. Withthe advent of machines that perform computations in the last century, the question ofrequired resources to solve a computational problem has become even more relevant.
Although there is some previous work, the systematic study of the resources requiredto solve a problem began in the 1960s. The area of computational complexity theory fo-cuses on problems rather than algorithms and its main goal is to determine the intrinsiccomplexity of computational problems. Problems are classified by the computational re-sources needed by an algorithm to solve them, for example the required time or space.The goal is to distinguish problems that can be solved in reasonable time or space, calledtractable problems, from problems that use too many resources, called intractable prob-lems.
Many problems that are studied in complexity theory are either graph problems orcan be rephrased in terms of graphs. Graphs are mathematical objects used to modelthe relations between entities, for instance the ‘friendship’ relation in a social network orthe structure of molecules in chemistry. A graph problem is then a well-defined questionabout graphs. This dissertation is concerned with graph problems.
The (time) complexity of a problem is the number of computational steps required bythe best known algorithm to solve the problem, measured as a function of the size of the
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input. To give an example, a graph with n vertices can be represented as a table with anentry for every pair of vertices, where every field tells us if a pair of vertices is adjacent.Such a table has n2 entries, and that is the size of the input of a graph problem.
A problem is usually considered to be tractable if the time complexity is bounded bya polynomial. The class of all problems that can be solved in polynomial time is calledP. It is harder to say when a problem is intractable. Researchers in complexity theorylack the tools to show lower bounds on problems, i.e. we do not know how to show thata problem is not contained in P. In the absence of provable lower bounds, a problem isusually considered to be intractable if a problem that we think is hard can be reducedto it (that is, a solution to the problem we want to show is hard can be used to solve aknown hard problem). The most prominent example is the class of NP-hard problems.
The class NP is the class of problems where, given a solution to the problem, we canverify the solution in polynomial time. A problem is NP-hard if it is at least as hard asall the problems in NP. Clearly P is a subclass of NP (if we can find a solution, we canalso verify it). The question whether P is a proper subclass of NP, i.e. if P 6= NP, is oneof the major open questions in complexity theory (and mathematics1 as a whole) sincethe 1970s.
Problems that are contained in NP (i.e. they have an easily checkable solution) andthat are also NP-hard (i.e. an algorithm solving them is powerful enough to solve allproblems in NP) are called NP-complete. Many graph problems have been shown tobe NP-complete and only exponential algorithms are known that solve them in general.However, many such problems are important and still have to be solved in practice, forinstance in the areas of computational biology or the study of social networks.
Moreover, the distinction between problems that are in P and problems that are NP-complete does not seem to fully answer our question above: what problems can we expectto solve using computers? Practitioners still succeed in solving NP-complete problems inpractice. For example, algorithms for the boolean satisfiability problem (SAT) continueto improve, and researchers submit their implementations to competitions that solveinstances in reasonable time. Does that mean that the theory is irrelevant? No, it justmeans that the instances that are solved in practice are very different from the worst caseand classical complexity theory does not address this issue well.
A more fine-grained study of computational complexity, introduced in the 1990s byDowney and Fellows, helps address these issues. In parameterized complexity theory, weare not just looking at the runtime of an algorithm as a function of the size of the input,but instead we also take other ‘parameters’ of the problem into account. As an illustration,the SAT problem becomes easy if we have just a few variables, or just a few clauses. Thisis what parameterized complexity tries to capture. A problem is said to be fixed-parametertractable if it can be solved in time O(f(k) · p(n)) for some computable function f (ofthe parameter k) and polynomial p (of the size of the input n). The SAT problem isfixed-parameter tractable parameterized by the number of variables in the formula, andalso by the number of clauses. This theory gives a good explanation of why some booleansatisfiability problems can be solved in practice: if the formulas were written by a human(or are derived directly from a human specification in some way), then one would expecta bias in these formulas to use few variables or few clauses. Thus it makes sense that inpractice many instances of the SAT problem are solvable in reasonable time.
1It is one of the famous Millenium Problems and there is a $1,000,000 prize from the Clay Mathematicalinstitute for solving it: http://www.claymath.org/millennium-problems/p-vs-np-problem
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Parameterized complexity theory also comes with its own notion of hardness. Justas in classical complexity theory there are no tools to prove that a problem is not fixed-parameter tractable. Instead a problem is considered to be intractable if it is shown tobe W[1]-hard.
The parameters are supposed to capture what is hard about a problem, and theintuition is that a problem is fixed-parameter tractable if it becomes tractable for smallvalues of the parameter. Of course this is very valuable for practical applications, wherepractitioners can take this into account to design algorithms that exploit small values ofthe parameter. But it is also very interesting for computer science theory: it gives us anidea of what, fundamentally, makes the problem hard.
How do we choose the parameters? The parameters should be broad enough andrelevant for applications, while still allowing many problems to become tractable. Apopular choice of parameter is just the size of the output. For instance, if we are lookingfor a vertex cover in a graph, it would be reasonable to expect that the problem is easier ifwe can find a small vertex cover. This is sometimes called the ‘natural parameterization’.
Another approach used for graph problems is to take the structure of the input graphinto account. Many NP-complete graph problems become tractable on classes of sparsegraphs, such as planar graphs or graphs of bounded tree-width. Tree-width is a measureof how far a graph is from being a tree and is in fact a widely used parameter in the studyof parameterized graph problems.
In this dissertation we identify a particularly interesting group of structural graphparameters that we refer to as distance parameters. They measure how far a graphis from belonging to some class of graphs. To give an example, the so called ‘VertexDeletion’ problems ask whether we can remove k vertices from a given graph to place thegraph in some class C, where k is the parameter. Many graph problems are in fact of thatform, even if it is not immediately apparent: the vertex cover number of a graph is justthe minimum number of vertices we have to delete from the graph in order to obtain anedgeless graph. The feedback vertex set number is just the minimum number of verticeswe have to delete from the graph in order to obtain a forest. The graph bipartizationproblem asks to delete the minimal amount of vertices to obtain a bipartite graph. Moregenerally, Cai [32] studies the parameterized complexity of distance measures defined interms of addition and deletion of vertices and edges to hereditary classes C, i.e. graphclasses that are closed under induced subgraphs.
But the notion of distance can be interpreted much more widely. Counting deletionsof vertices and edges gives a rather simple notion of distance, and many more involvednotions have also been studied. For example the popular parameter tree-width. Anotherclassic example is the crossing number of a graph, which provides a notion of distance tothe class of planar graphs.
A common motivation for studying distance parameters is the following: if a problemone wishes to solve is known to be tractable on a class of graphs C, then it is reasonable toconjecture that the problem might also be tractable for graphs that are ‘almost’ in C, orclose to C in some sense. Thus the distance to C provides an interesting parameterizationof that problem (called distance to triviality by Guo et al. [95]). Since many problemsare tractable on classes of sparse graphs, distances to sparse graphs are a good target forthis.
Take for example the Vertex Cover problem, one of the most studied problemsin parameterized complexity theory. It can be solved in polynomial time on the class of
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bipartite graphs, and on trees. If we consider the deletion distance to trees as a parameter,then it is fixed-parameter tractable – one way to see this is by considering that the tree-width of a graph is in this case bounded by the parameter, and Vertex Cover isknown to be FPT parameterized by tree-width [17]. Other successful examples of thisstrategy include the study of modulators to graphs of bounded tree-width in the contextof kernelization (see [73, 78]), or the parameterizations of colouring problems (see [121]).In the context of satisfiability problems, this strategy has lead so-called backdoors[81, 83];parameterizations by the number of variables that need to be fixed to place the problemin a class that can be solved in polynomial time.
Another topic of research is the relationship between parameters. The study of the re-lationship between parameters leads to a better understanding of the complexity landscapeand generally offers a good way to discover new tractable or intractable parameterizationsof problems. Figure 1.1 shows which of the distance parameters in this dissertation providean upper bound for other graph parameters. For many problems it is unknown for ex-actly which parameters they are tractable. For example, it might be known that the graphproblem is fixed-parameter tractable parameterized by vertex cover number, and that itis W[1]−hard parameterized by tree-width. But its complexity status for the parameterstree-depth or path-width might be unknown. A better understanding of the relationshipbetween the parameters allows us to ask the right questions, which might eventually leadus to close these gaps and draw the line between tractability and intractability in a moreprecise way.
In this dissertation we make contributions to the study of distance parameters inseveral ways. In Chapter 4 we discuss the distance parameter tree-depth, which is locatedin-between vertex cover number and tree-width. We establish several fixed-parametertractability and hardness results for problems parameterized by tree-depth, closing somegaps in the knowledge for these problems.
In Chapter 5 we introduce a new parameter, elimination distance to a class C, thatis a natural generalisation of both deletion distance to a class C and tree-depth. Just asdeletion distance is defined with respect to some class C, it is defined as the eliminationdistance to some class C. Recall that vertex cover number is the deletion distance tothe class of edgeless graphs. Similarly, we show in Section 5.4 that tree-depth can beseen to be the elimination distance to the class of edgeless graphs. Indeed, eliminationdistance generalises deletion distance in the same way that tree-depth generalises vertexcover number.
Broadly speaking, there are two kinds of questions one can ask about a graph param-eter: Given a graph G, is it FPT to determine the value of the parameter for the graph,or, in other words, is it FPT to check whether the graph has parameter at most k pa-rameterized by k? The second kind of question is: What kind of graph problems becometractable when parameterized by the parameter? We discuss both these questions forelimination distance.
We show an application of the parameter in Chapter 6, where we demonstrate thatGraph Isomorphism is FPT parameterized by elimination distance to bounded degree.This generalises a result of Bouland et al. [23] and is an example of the second kind ofquestion.
In Chapter 7 we give an answer to a question of the first kind. We show that giventhe finite set of minors that characterize a minor-closed class of graphs C, we can findthe minors that characterize the graphs with elimination distance k to C. So if the set of
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vertex cover number
tree-depth
path-width
tree-width
degeneracy
max leaf number
generalised tree-depth
deletion distance to bounded degree
elimination distance to bounded degree
Figure 1.1: A diagram that illustrates the relationships between distance parameters to sparsegraph classes that are part of this dissertation. An arrow between two parameters should beinterpreted as ‘a function of the first parameter gives an upper bound for the second parameter’.
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excluded minors is available for a minor-closed class C, there is an explicit FPT algorithmto test for membership in the class of graphs with elimination distance k to C.
Algorithmic meta-theorems are algorithmic results that apply not only to a singleproblem, but to a whole class of problems. A seminal result in the area is Courcelle’sTheorem, which states that every problem definable in monadic second-order logic can bedecided in linear time on bounded tree-width2. Our result in Chapter 7 is an algorithmicmeta-theorem because it applies to all minor-closed graph classes.
In Chapter 8, building on recent work of Grohe et al. [92], we establish anotheralgorithmic meta-theorem. We show that every graph problem that is both slicewisenowhere dense and slicewise first-order definable (these terms are defined precisely inDefinition 8.2.1 and Definition 8.2.2) is FPT.
Lastly, in Chapter 9 we discuss two problems that have natural distance parameters.We parameterize by these parameters, and also consider additional structural distanceparameters. We establish fixed-parameter tractability results for these parameterizations.
1.1 Related areas of research
Here we give an overview of the different branches of mathematics and theoretical com-puter science that are related to the research of this dissertation. More specific relatedwork is mentioned in the individual chapters.
Parameterized complexity theory. This dissertation explores the parameterized com-plexity of several problems with regard to distance parameters. Parameterized complexitywas introduced in the 1990s by Downey and Fellows [56] and extends the classical (one-dimensional) complexity theory to a two-dimensional theory that measures the complexityof a problem not only as a function of the size of the input, but also in terms of an addi-tional parameter.
The general aim is a more fine-grained analysis of (NP-)hard problems, gaining abetter understanding of tractable instances. There is a large amount of related work inthat area, which is referred to in the individual chapters.
We formally introduce the theory and state all the important definitions in Section 2.2.
Graph editing: vertex and edge deletion/addition problems. A natural notionof distance of a graph to a class of graphs comes from editing operations on a graph. Howmany vertex or edge deletions or additions do we have to perform on a graph G to put itinto some class C?
In the 1980s Lewis and Yannakakis [112] showed that vertex deletion problems areNP-hard for any hereditary non-trivial classes, and Yannakakis [159] showed that edgedeletion problems are NP-hard for several graph classes.
These hardness results made these problems a natural candidate for the study of theirparameterized complexity. For this reason researchers in the 1990s focused on showingtractability or hardness of such problems within the framework of parameterized com-plexity.
Kaplan et al. [102] showed the tractability of parameterized completion problems onchordal, strongly chordal, and proper interval graphs. Cai [32] showed that the deletion
2The statement can in fact be extended to clique-width; see Section 3.5.
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distance problem to any hereditary property that is characterised by a finite set of for-bidden subgraphs is fixed-parameter tractable (also see below). Cai [33] also showed thatGraph Colouring is FPT parameterized by the some edit distances to split graphsand bipartite graphs. Mathieson [123] studies the parameterized complexity of a range ofgraph editing problems in his doctoral dissertation, where the graphs are edited to satisfya variety of degree constraints.
It is also possible to go one step further and consider the structure of the subgraphinduced by the deleted vertices. A set of vertices that, if removed from a graph, places itin a certain class of graphs is sometimes also called a modulator. Instead of consideringthe size, one can also consider the structure of the modulators . [58, 82].
Finite model theory. Model theory is the study of mathematical structures (e.g.graphs) from the perspective of logic.
Finite model theory is a branch of model theory that only deals with finite structures,i.e. structures over a finite universe. Many important theorems, e.g. the compactnesstheorem, do only hold for infinite structures, so finite model theory is quite different fromgeneral model theory that also allows infinite structures. The objects studied in computerscience are usually finite, so it found many applications here: it is used in descriptivecomplexity theory, database theory and formal language theory. Most of the research inthe area is about the expressive power of logics on classes of structures. Libkin [113] givesa good introduction to the area in his book.
In Chapter 8 we use model theoretic language to study problems that are (slicewise)definable in first-order logic and slicewise nowhere dense.
Descriptive complexity theory. Descriptive complexity theory is concerned with theapplication of logic to questions in complexity theory. The main driving question is theapplication of logic to the P vs NP question by attempting to solve the correspondingquestion in model theory: Is there a logic that captures P and a logic that captures NP,and a way to show that these logics are different?
An important result in the area is Fagin’s theorem [61], first proved in his PhD disser-tation, which states that the class NP is exactly the set of languages that can be definedusing existential second-order logic. But while there is a logic that captures NP, a logicthat captures P continues to be elusive. However, the search for such a logic has spawneda tremendous amount of research. Most results in descriptive complexity are about char-acterising complexity classes by the logics that can express the problems in them; see thebook of Immerman [100] for a good overview.
Algorithmic meta-theorems. Descriptive complexity theory spawned another sub-field, often called algorithmic meta-theorems. It (mostly) studies the computational com-plexity of the model checking problem on different classes of structures, usually graphs.Results in the area are called algorithmic meta-theorems, because they yield algorithmicresults that (instead of just a single problem) apply to a whole class of problems.
The first main result in the area was Courcelle’s Theorem [39], which showed that allgraph properties definable in monadic second-order logic can be decided in linear timeon graphs of bounded tree-width. In the last 20 years there has been ample research onsimilar questions and it is now an established area, closely related to finite model theoryand descriptive complexity [91, 109].
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Although most results in the area are related to evaluating formulas from different log-ics, there are also meta-theorems unrelated to logic. For example the result of Cai [32] thatthe deletion distance problem to any hereditary property that is characterised by a finiteset of forbidden subgraphs is fixed-parameter tractable can be viewed as an algorithmicmeta-theorem.
A major quest over the last 20 years was to pin down the graph classes (closed undertaking subgraphs) for which first-order model checking is FPT. It is well known thatevaluating first-order formulas on general structures is PSPACE-hard. This holds evenon structures with only two elements, which is why structural restrictions alone will notreduce the complexity. However, the tools from parameterized complexity allow for a fineranalysis. The general problem of first-order model checking parameterized by the size ofthe formula is in XP. So the interesting question is: for which graph classes is it FPT?
Showing that first-order model checking is FPT on a class of graphs gives us an al-gorithmic meta-theorem in the following way: if a parameterized problem is slicewisefirst-order definable (i.e. the size of the formula depends on the parameter), then theproblem becomes fixed-parameter tractable on that class. For instance the independentset problem parameterized by the size of the independent set can be translated into afirst-order formula with size bounded in the parameter; the formula for the kth slice isthe following:
∃x1 . . . ∃xk.∧i<j
¬E(xi, xj).
The question has been researched for the last two decades. In particular, there hasbeen important research on proving such results for classes of sparse graphs in the recentpast. Courcelle’s result on monadic second-order logic was later generalised to clique-width/rank-width [42] and there is reason to believe it can not be pushed any further. Formany sparse graph classes, i.e. where the number of edges as a function of the minimumnumber of vertices grows only slowly, the problem of evaluating first-order sentences in theclass becomes fixed-parameter tractable. Seese [147] showed this for the class of graphsof bounded degree. Frick and Grohe [76] succeeded in generalising this further, to theclass of graphs of bounded local tree-width, which includes the classes of bounded tree-width, bounded degree and planar graphs. Flum and Grohe [71] proved that evaluating afirst-order sentence is also fixed-parameter tractable for the class of graphs that excludea minor. Dawar, Grohe and Kreutzer [50] further generalised this result to the class ofgraphs that locally exclude a minor, which includes the class of graphs of bounded localtree-width and also the class of graphs that exclude a minor. The result by Dvorak, Kraland Thomas [57] generalises this even more, by describing a linear time algorithm forevaluating first-order sentences on graphs of (locally) bounded expansion, which includesgraphs that (locally) exclude minors. The latest result by Grohe et al. [92] likely pushedthis to its limit by showing that first-order logic is FPT on nowhere dense classes of graphs.Kreutzer and Dawar [53] showed that first-order logic is not fixed-parameter tractable onsomewhere dense classes of graphs (unless FPT = AW[∗]).
So there has been tremendous progress over the years in generalising the result tomore and more general classes of sparse graphs.3 See Figure 1.2 for an illustration of the
3It should be remarked that sparse graphs are not the only graphs for which first-order model checkingis fixed-parameter tractable. For example the class of graphs of bounded clique-width mentioned aboveis not sparse, but does allow for efficient evaluation of first-order sentences [42].
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excluded minors bounded local treewidth
planar graphs bounded degree
bounded expansion locally excluded minors
acyclic graphs
bounded genus
bounded treewidth
nowhere dense
Figure 1.2: An illustration of sparse graph classes and their inclusion relation (taken from[48]).
relationships between some sparse graph classes.In Chapter 8 we extend the result by Grohe et al. [92] to show that any graph property
that is slicewise first-order definable and slicewise nowhere dense (these terms are definedprecisely in Definition 8.2.1 and Definition 8.2.2) is decidable in FPT time.
1.2 Contribution
This section gives an overview of the chapters, detailing our contribution in each of thechapters.
1.2.1 Chapter overview
Chapter 2 In order to provide the necessary background for the rest of the disserta-tion, we introduce the basic notions of first-order model theory, parameterized complexitytheory, some areas of graph theory, and order theory.
Chapter 3 This chapter provides additional background. It introduces the notion ofa distance parameter, defined to be a parameter that measures how far a graph is frombeing contained in some class. It then discusses several structural graph parameters thatcan be understood as distance parameters.
Chapter 4 This chapter is concerned with the distance parameter tree-depth and ex-hibits several results about it. We prove fixed-parameter tractability and hardness resultsfor a number of problems parameterized by tree-depth.
Chapter 5 In this chapter we introduce the new graph parameter elimination distanceto a class C. The parameter naturally generalises the notion of deletion in deletion dis-
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tance, and we prove that tree-depth is just the elimination distance to the class of edgelessgraphs. We show that the parameter offers a range of algorithmically interesting charac-terisations.
Chapter 6 In this chapter we give an application of the parameter elimination distanceintroduced in Chapter 5. We discuss the Graph Isomorphism problem and establishthat Graph Canonisation, and thus Graph Isomorphism, is FPT when parameter-ized by elimination distance to bounded degree, extending results of Bouland et al. [23].
Chapter 7 This chapter discusses another application of the parameter eliminationdistance introduced in Chapter 5. We prove an algorithmic meta-theorem, establishingthat the problem of determining elimination distance to any minor-closed class C is FPT.We show that we can provide an explicit FPT algorithm if a set of forbidden minorscharacterising C is given.
Chapter 8 In this chapter we prove another algorithmic meta-theorem. We establishthat any problem that is both slicewise nowhere dense and slicewise first-order definable(these terms are defined in Definition 8.2.1 and Definition 8.2.2) is FPT.
Chapter 9 In this chapter we discuss two graph problems parameterized by their nat-ural distance parameters. First we show that Clique Dominating Set is FPT onnowhere dense classes of graphs. Next, we consider the Anchored k-core problem andshow that it is FPT parameterized by tree-width and budget, as well as degree, size of thecore and clique-width.
Chapter 10 Here we give a final discussion of the results established in this dissertation,and discuss a wide range of potential further research directions.
1.2.2 Contribution details
The content of Chapter 5 and Chapter 6 is based on [28] and [30], both being coau-thored with Anuj Dawar. The content of Chapter 7 and Chapter 8 is based on [29], alsocoauthored with Anuj Dawar.
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CHAPTER 2
General background
In order to provide the necessary background for the rest of this dissertation, we hereintroduce the basic notions of first-order model theory, parameterized complexity theory,some areas of graph theory and order theory.
We introduce model theory in Section 2.1, giving a brief overview of first-order logicand second-order logic. Next in Section 2.2 we introduce the elementary notions of pa-rameterized complexity theory, briefly mentioning kernelizations and an equivalent for-malisation using model theoretic notions. In Section 2.3, we give a brief introduction ofgraph theory and discuss the basics of graph minor theory, nowhere dense graph classesand graph classes with bounded expansion, as well as the the graph isomorphism problem.Lastly we introduce some basic order theoretic notions in Section 2.4.
In Chapter 3 we introduce more specific background: there we motivate the notionof distance parameters, i.e. structural graph parameters that measure how far a graph isfrom being contained in some class of graphs, and introduce several such parameters.
2.1 Model theory
Here we introduce the elementary notions of model theory, roughly following the book byHodges [98].
The core notion of model theory is that of a structure, and based on that we definesubstructures, expansions and reducts.
Definition 2.1.1. A (relational) signature or vocabulary σ is a finite set of relationsymbols, each with an associated non-negative integer, called its arity, and a finite set ofconstant symbols.
Definition 2.1.2. A σ-structure A consists of a set V (A) called the universe of A andfor each k-ary relation symbol R ∈ σ a relation R(A) ⊆ V (A)k, and for each constantsymbol c ∈ σ, an element c(A) ∈ V (A). A σ-structure A is called finite if its universe isfinite.
Remark. Our structures are mostly (coloured) graphs, so σ = E or σ = E,C1, C2, . . . , Crwhere E is the (binary) edge relation and the Ci are unary relation symbols and we donot have any constants. A graph G is then a finite σ-structure with vertex set V (G), edgerelation E(G), and colours Ci(G).
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Considering just a part of the elements of a structure gives rise to substructures.
Definition 2.1.3. Let A be a σ-structure. For a subset of the universe V ′ ⊆ V (A), thesubstructure A[V ′] of A induced by V ′ is the structure with universe V (A[V ′]) = V ′ andfor each k-ary relation symbol R ∈ σ interpretation R(A[X]) = R(A) ∩ V ′k.
It is also possible to reduce and expand a structure.
Definition 2.1.4. Let σ be a signature. For a subset σ′ ⊆ σ, the σ′-reduct of a structureA is the σ′-structure A′ with V (A′) = V (A) and R′(A′) = R′(A) for all R′ ∈ σ′.
Whenever a σ′-structure A′ is a σ′-restriction of a σ-structure A, we say that A is aσ-expansion of A′.
2.1.1 First-order logic
In this section we review the syntax and semantics of first-order logic.The main notion here is that of a formula, and terms are the basic building blocks of
formulas.In the following we assume a given (countably infinite) set of variables, usually denoted
by x, y or x1, x2, . . . .
Definition 2.1.5. A σ-term is either a variable or a constant from σ.
Definition 2.1.6. A σ-formula is defined as follows.
• IfR is an n-ary relation symbol from σ and t, u, t1, . . . , tn are terms, thenR(t1, . . . , tn)and t = u are σ-formulas. They are said to be atomic.
• If ϕ, ψ are σ-formulas and x is a variable, then ¬ϕ, ϕ ∧ ψ and ∃xϕ are σ-formulas.
We also define the following abbreviations:
Definition 2.1.7. Let ϕ and ψ be σ-formulas. Then we define:
• ϕ ∨ ψ := ¬(¬ϕ ∧ ¬ψ);
• ϕ→ ψ := ¬ϕ ∨ ψ;
• ϕ↔ ψ := (ϕ→ ψ) ∧ (ψ → ϕ);
• ∀xϕ := ¬∃x¬ϕ;
• > := ∀x(x = x)
• ⊥ := ¬>.
Definition 2.1.8. The subformulas of a formula ϕ are defined as follows.
• If ϕ is atomic, then ϕ is the only subformula.
• If ϕ is of the form ¬ψ or ∃xψ, then ϕ and the subformulas of ψ are subformulas ofϕ
• If ϕ is of the form ψ1∧ψ2, then ϕ and the subformulas of ψ1 and ψ2 are subformulasof ϕ.
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Definition 2.1.9. An occurrence of a variable x is bound if it is in a subformula of theform ∃xϕ. Otherwise the occurrence is free. The free variables of a formula are thevariables with free occurrences. A formula with no free variables is called a sentence. Atheory is a set of sentences.
Definition 2.1.10. The quantifier rank of a formula ϕ is the nesting depth of quantifiersin ϕ.
2.1.2 Second-order logic
Second-order logic is an extension of first-order logic that also allows quantification overrelations. In addition to variables x that can range over elements of the universe, we alsoassume the existence of a given (countably infinite) set of variables R that can range overrelations.
The fragment of second-order logic that only allows quantification over unary relations,i.e. sets, is called monadic second-order logic (or just MSO).
Following the convention in Flum and Grohe’s book [72], we call a second-order formulawhere all relation variables occur freely a first-order formula with free relation variables.
Next we define some more useful notation.
Definition 2.1.11. Let A be a structure. Let ϕ(X) be a first-order formula with exactlyone free relation variable X of arity s. We say that a set S ⊆ As is a solution for ϕ ifA |= ϕ(S).
In the following we (inductively) define two families of classes of formulas Σi and Πi
that are needed to define the parameterized complexity classes W[i] in Section 2.2.3.
Definition 2.1.12. We let Σ0 and Π0 both denote the class of all quantifier-free formulas.For i > 0, we let Σi be the class of all second-order formulas of the form
∃x1 . . . ∃xk.ψ
for all k ∈ N and ψ ∈ Πi−1, and let Πi be the class of all second-order formulas of theform
∀x1 . . . ∀xk.ψ
for all k ∈ N and ψ ∈ Σi−1.
2.2 Parameterized complexity theory
Parameterized complexity theory is a two-dimensional approach to the study of the com-plexity of computational problems. A thorough discussion of the subject can be found inthe books by Downey and Fellows [56], Flum and Grohe [72] and Niedermeier [135]. Wefollow notation and terminology from [72].
Here we introduce the basic definitions of the theory. In Section 2.2.4 we also give non-standard definitions of parameterized problems and fixed-parameter tractability based onmodel theoretic notions. These are more convenient for our study of slicewise nowheredense and slicewise first-order definable problems in Chapter 8.
The notions of language and problem are standard notions within complexity theory.
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Definition 2.2.1. Let Σ be a finite alphabet. A language (or problem) L is a set ofstrings L ⊆ Σ∗.
The main addition of parameterized complexity theory is the association of an addi-tional parameter with each problem instance.
Definition 2.2.2. Let Σ be a finite alphabet. A parameterization is a computable func-tion κ : Σ∗ → N. For a parameterization κ and a problem L ⊆ Σ∗, we say that (L, κ) isa parameterized problem over Σ.
2.2.1 Fixed-parameter tractability
The central notion of parameterized complexity is that of fixed-parameter tractability,which captures the idea of tractability within the framework. If we can decide a problemin polynomial time and the degree of the polynomial is fixed and independent of theparameter, then the problem is fixed-parameter tractable (even though there might be afactor of an arbitrary computable function of the parameter involved).
Definition 2.2.3. Let Σ be a finite alphabet. We say that L is fixed-parameter tractablewith respect to a parameterization κ if we can decide whether an input x ∈ Σ∗ is in Lin time O(f(κ(x)) · |x|c), where c is a constant and f is some computable function. Theclass of all fixed-parameter tractable problems is called FPT.
Remark. Although FPT is a class of problems, the convention is to say a problem is FPTinstead of saying it is in FPT. If the algorithm decides the problem in time O(f(κ(x))·|x|),we say that the problem is (in) linear FPT.
Remark. To emphasise the parameter dependence, we sometimes write O∗(f(k)) insteadof O(f(k) · nc), for some computable function f and constant c.
2.2.2 Kernelization
Central to the study of parameterized complexity of computational problems is an alter-native characterisation of fixed-parameter tractability called kernelization. Kernelizationcaptures the idea of pre-computation: a problem has a kernel if we can reduce the probleminstance to a smaller one (i.e. bounded in size by a function of the parameter).
Definition 2.2.4. Let Σ be a finite alphabet and let (L, κ) be a parameterized problemover Σ. A polynomial-time computable function K : Σ∗ → Σ∗ is a kernelization of (L, κ)if there exists a computable function h : N→ N such that for all x ∈ Σ∗ we have that
• x ∈ L ⇐⇒ K(x) ∈ L; and
• |K(x)| ≤ h(κ(x)).
The instance K(x) is called the kernel of x ∈ Σ∗.
It is straightforward to show that a parameterized problem admits a kernelization if,and only if, it is fixed-parameter tractable. (See [72, Theorem 1.39] for a proof.)
Theorem 2.2.5. A decidable parameterized problem (L, κ) over a finite alphabet Σ isfixed-parameter tractable if, and only if, it has a kernelization.
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Kernelizations give a different perspective on fixed-parameter tractability. We canobtain a more fine-grained understanding of the complexity of a problem by consideringthe size of the kernel. A large amount of current research is dedicated to showing not onlythe existence of a kernelization for a specific problem, but to showing the existence of asmall kernelization, i.e. a kernelization that has its size bounded by a polynomial functionof the parameter, or even a linear one.
2.2.3 The W[i] hierarchy and the class XP
Parameterized complexity theory also has a notion of intractability. In classical complexitytheory we consider a problem to be intractable if it is not contained in the class P. How-ever, we cannot prove this for problems, and the usual way to demonstrate intractabilityis by showing that a problem is NP-hard. Most complexity theorists believe that P 6= NP,so it is generally thought to be unlikely that an NP-hard problem is contained in P.
The situation in parameterized complexity mirrors that in classical complexity the-ory. The notion of tractability in parameterized complexity is that of fixed-parametertractability and FPT is the class of such problems. There is also a notion of intractabilitythat plays a role comparable to NP. However, there is not a single class, but a wholehierarchy: the classes W[i] for i > 0. On top of those classes is the class XP that con-tains all of them. Sometimes just the first of these classes W[1] is considered to be theparameterized version of NP.
The W[i] classes were originally defined by Downey and Fellows [56] using notionsfrom circuit complexity; each class is based on a circuit satisfiability problem. We givean equivalent definition here in terms of logic.
At the core is the Weighted Fagin Definability problem for ϕ defined asfollows. Let ϕ(X) be a formula with one free relation variable X of arity s.
Weighted Fagin Definability Problem for ϕ (WDϕ)Input: A structure A and a non-negative integer kParameter: kProblem: Is there a solution S ⊆ As for ϕ of cardinality |S| = k?
Then the W-hierarchy is defined as follows. Recall from Definition 2.1.11 the notionof a solution of a formula, and from Definition 2.1.12 the class of formulas Πi. We write[·]fpt to denote the closure of a class under FPT-reductions.
Definition 2.2.6. For every i > 0 we define
W[i] := [WDϕ | ϕ(X) ∈ Πt]fpt
Next we define the class XP that subsumes all the previously defined ones. It issometimes compared to the class PSPACE in classical complexity theory, although it isnot defined in terms of space resources. The class XP contains all the parameterizedproblems where the growth of the polynomial is allowed to depend on the parameter.
Definition 2.2.7. Let Σ be a finite alphabet. The class of parameterized problems XPcontains all the parameterized problems L with parameterization κ that can be decidedin time O(nf(κ(x))) for some computable function f .
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Without proof we state the following inclusions between the classes defined here (seee.g. [72, Chapter 7]):
FPT ⊆ W[x1] ⊆ W[2] ⊆ · · · ⊆ XP
2.2.4 Model theoretic definitions
We find it convenient in same parts of the dissertation to not define parameterized prob-lems as classes of strings, but as classes of structures. This avoids the need to deal withparticularities of the string encoding. The idea of defining problems this way is inspiredby Dawar and He [51]. We introduce the definitions in this section.
We always explicitly mention when we refer to these non-standard definitions insteadof the ones given above.
Definition 2.2.8. Let σ be a signature. A problem Q ⊆ str(σ) is an (isomorphism-closed)class of σ-structures.
Definition 2.2.9. Let σ be a signature. A parameterization is a computable functionκ : str(σ)→ N.
Definition 2.2.10. Let σ be a signature and let Q be a problem. We say that Q isfixed-parameter tractable with respect to a parameterization κ if we can decide whetheran input A ∈ str(σ) is in Q in time O(f(κ(A)) · |A|c), where c is a constant and f is somecomputable function.
2.3 Graph theory
Here we introduce the elementary notions of graph theory, graph minor theory, the iso-morphism problem, and certain sparse graph classes.
A good introductory text to the topic is the book of Diestel [55], and for sparse graphclasses the book of Nesetril and Ossona de Mendez [133]. Our notation is mostly basedon these books.
Definition 2.3.1. A graph G is a set of vertices V (G) and a set of edges E(G) ⊆V (G)×V (G). We write |G| for the number of vertices |V (G)| and ||G|| for the cardinalityof the edge set |E(G)|.
Remark. We usually assume that graphs are loop-free and undirected, i.e. that E is ir-reflexive and symmetric. If E is not symmetric, we call G a directed graph.
Remark. There is also an obvious connection to model theory discussed above: A graph isa relational structure with σ = E – a signature that only contains one binary relation,the edge relation. If σ also contains unary relations the structure can be conceptualizedas a coloured graph.
A basic notion of graph theory is that of a neighbour, and the number of neighbours– the degree.
Definition 2.3.2. Let G be a graph. The neighbourhood of a vertex v is NG(v) := w ∈V (G) | vw ∈ E(G). For a set of vertices S ⊆ V (G) its neighbourhood is defined to beNG(S) :=
⋃v∈S NG(v) \ S.
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Definition 2.3.3. Let G be a graph. If v ∈ G and S ⊆ V (G), we write EG(v, S) for theset of edges vw | w ∈ S between v and S.
Definition 2.3.4. Let G be a graph. The degree of a vertex v ∈ V (G) is the size of itsneighbourhood degG(v) := |NG(v)|. If it is clear from the context what the graph is, wemay omit the subscript. The maximum degree of a graph G is the maximum degree of itsvertices ∆(G) := maxdegG(v) | v ∈ V (G).
Just as for structures in general, we can think of induced subgraphs as induced sub-structures. We also define the more general notion of a subgraph that might inherit onlysome of the edges from the original graph.
Definition 2.3.5. Let G be a graph. A subgraph H of G is a graph with vertices V (H) ⊆V (G) and edges E(H) ⊆ (V (H) × V (H)) ∩ E(G). If A ⊆ V (G) is a set of verticesof G, we write G[A] for the subgraph induced by A, i.e. V (G[A]) = A and E(G[A]) =E(G) ∩ (A× A).
If A is a subset of V (G), we write G \ A for G[V (G) \ A]. For a vertex v ∈ V (G), wewrite G \ v for G \ v.
Graphs can be thought of as networks, where the vertices are connected by strings ofedges – so it is natural to define reachability, paths and connectedness.
Definition 2.3.6. Let G be a graph. A vertex v ∈ V (G) is said to be reachable from avertex w in G if v = w or if there is a sequence of edges a1a2, . . . , as−1as ∈ E(V ) with theai pairwise distinct and w = a1 and v = as. We call the subgraph P of G with verticesV (P ) = a1, . . . , as and edges E(P ) = a1a2, . . . , as−1as a path from w to v.
Definition 2.3.7. Let H be a subgraph of a graph G and v, w ∈ V (G). A path through Hfrom w to v is a path P from w to v in G with all vertices, except possibly the endpoints,in V (H), i.e. (V (P ) \ v, w) ⊆ V (H).
This notion also gives rise to an extended definition of neighbourhood.
Definition 2.3.8. Let G be a graph. The r-neighbourhood of a vertex v ∈ V (G) is
Nr(v) := w ∈ V (G) | there is a path from v to w of length at most r.
It is easy to see that for undirected graphs reachability defines an equivalence relationon the vertices of G.
Definition 2.3.9. A subgraph of an undirected graph induced by a reachability class iscalled a (connected) component.
Definition 2.3.10. Let r ∈ N. An r-independent set in a graph G is a set of vertices ofG such that their pairwise distance is at least r.
Definition 2.3.11. Let G be a connected graph. The eccentricity of a vertex v ∈ V (G)is the maximum length of a shortest path between v and any other vertex u ∈ V (G). Theradius of G is the minimum eccentricity of all vertices in G. The diameter of G is themaximum eccentricity of all vertices in G.
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2.3.1 Graph isomorphism
The graph isomorphism problem is the question whether two graphs are structurallyequivalent, or more precisely, given two graphs if there is an edge relation preservingbijection between them. One of the major questions in complexity theory is whether itcan be decided in polynomial time. A detailed discussion of the problem can be found inChapter 6, which is dedicated to the problem.
We assume in the following that a graph G is given by its adjacency matrix, so thatevery vertex of G naturally corresponds to a number between 0 and |G| − 1.
Formally, the notion of isomorphic graphs is defined as follows:
Definition 2.3.12. Two graphs G, G′ are isomorphic if there is a bijection ϕ : V (G)→V (G′) such that for all v, w ∈ V (G) we have that vw ∈ E(G) if, and only if, ϕ(v)ϕ(w) ∈E(G′). We write G ∼= G′ if G and G′ are isomorphic. We write GI to denote the problemof deciding, given G and G′, whether G ∼= G′.
Definition 2.3.13. A (k-)colouring of a graph G is a map c : V (G) → 1, . . . , k forsome k ∈ N. We call a graph together with a colouring a coloured graph.
If no two adjacent vertices share the same colour, we say the colouring is proper.
Definition 2.3.14. A colouring c : V (G) → 1, . . . , k of a graph G is a proper (k-)colouring if c(u) 6= c(v) whenever uv ∈ E(G).
Definition 2.3.15. Two coloured graphs G,G′ with respective colourings c : V (G) →1, . . . , k, c′ : V (G′)→ 1, . . . , k are isomorphic if there is a bijection ϕ : V (G)→ V (G′)such that:
• for all v, w ∈ V (G) we have that vw ∈ E(G) if, and only if, ϕ(v)ϕ(w) ∈ E(G′);
• for all v ∈ V (G), we have that c(v) = c′(ϕ(v)).
Remark. Note that we require the colour classes to match exactly, and do not allow apermutation of the colour classes.
Definition 2.3.16. Let C be a class of (coloured) graphs closed under isomorphism. Acanonical form for C is a function F : C → C such that
• for all G ∈ C, we have that F (G) ∼= G;
• for all G,H ∈ C, we have that G ∼= H if, and only if, F (G) = F (H).
For the following definition we need the notion of a pre-order, defined below in Def-inition 2.4.1. Note that we can define a lexicographical ordering on graphs via theiradjacency matrices.
Definition 2.3.17. A canonical form F for a class C gives rise to a natural pre-order onC, which we denote vF whereby G vF H just in case F (G) is lexicographically smallerthan F (H). Then for any pair of graphs G and H in C, at least one of G vF H or H vF Gmust hold, and both hold if, and only if, G ∼= H.
2.3.2 Graph minor theory
A graph H is a minor of another graph G, if H can be obtained from G by repeatedlydeleting vertices and edges, and by contracting1 edges. Many of the important results in
1An edge uv is contracted by creating a new vertex w that shares the neighbours of both sides of theedge uv (i.e. N(w) = N(u) ∪N(v)) and then removing u and v from the graph.
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the area were developed by Robertson and Seymour in a series of twenty papers. Thissection introduces the basic notions of graph minor theory.
Instead of the definition above, we define the notion of minor in an equivalent2 wayusing the notion of a minor map. This definition makes it easier to talk about shallowminors later that only allow contractions in a limited radius.
Definition 2.3.18. A graph H is a minor of a graph G, written H G, if there is a map,called the minor map, that takes each vertex v ∈ V (H) to a tree Tv that is a subgraph ofG such that for any u 6= v the trees are disjoint, i.e. Tv ∩ Tu = ∅, and such that for everyedge uv ∈ E(H) there are vertices u′ ∈ Tu, v′ ∈ Tv with u′v′ ∈ E(G).
Definition 2.3.19. A class of graphs C is minor-closed if H G and G ∈ C impliesH ∈ C.
Definition 2.3.20. The set of minimal excluded minors M(C) is the set of graphs in thecomplement of C such that for each G ∈M(C) all proper minors of G are in C.
Robertson and Seymour [142] show that every minor-closed class is characterised bya finite set of ‘forbidden minors’.
Theorem 2.3.21 (Robertson-Seymour Theorem [142]). If a graph class C is minor-closed,then M(C) is finite.
Remark. It is an important consequence of this theorem that membership in a minor-closed class can be tested in polynomial time. This is because, as part of the proof,Robertson and Seymour [142] also show that we can test whether a graph is a minor ofanother graph in polynomial time.
Definition 2.3.22. For a set M of graphs, we write Forb(M) for the class of graphswhich forbid M as minors, i.e. Forb(M) = G | H 6 G for all H ∈M.
2.3.3 Nowhere dense classes of graphs and classes with boundedexpansion
In the following we use the language of minors to define nowhere dense classes of graphs.These are a very general notion of sparse graph classes that subsume many other sparsegraph classes.
For that we need the notion of a shallow minor:
Definition 2.3.23. Let r ∈ N. A minor H of G is a shallow minor of G of depth r orjust a depth-r minor of G, written H r G, if there is a minor map that takes verticesin H to trees that have radius at most r. We write G∇r for the set of shallow minors ofdepth r of G.
Definition 2.3.24. A class of graphs C is nowhere dense if for every r ∈ N there is agraph Hr such that Hr 6r G for all G ∈ C. A nowhere-dense class of graphs C is calledeffectively nowhere dense if there is a computable function h from integers to graphs suchthat if G ∈ C then for all r we have h(r) 6r G.
2See [55, Proposition 1.7.1] for a proof.
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Remark. We are only interested in effectively nowhere-dense classes so we simply use theterm nowhere dense to mean effectively nowhere dense.
Dawar [47, 49] introduced the notion of quasi-wide graph classes that was proved byNesetril and Ossona de Mendez [133] to be equivalent to that of nowhere-dense classes ofgraphs closed under induced subgraphs.
Definition 2.3.25. Let G be a graph and let r ≥ 0 be an integer. Then a set A ⊆ V (G)is r-scattered if for all u 6= v ∈ A, we have Nr(u) ∩Nr(v) = ∅.
Definition 2.3.26. Let s : N→ N be a function. A class C of graphs is quasi-wide withmargin s if for all r ≥ 0 and m ≥ 0 there exists an N ≥ 0 such that if G ∈ C and |G| > Nthen there is a set S ⊆ V (G) with |S| < s(r) such that G \ S contains an r-scattered setof size at least m.
Theorem 2.3.27 (Nesetril and Ossona de Mendez [133]). Let C be a class of graphs closedunder taking induced subgraphs. Then C is nowhere dense if, and only if, it is quasi-wide.
Nesetril and Ossona de Mendez also introduce a characterisation of nowhere densegraph classes using limits on the ratio of the number of edges to the number of vertices.They prove the following trichotomy:
Definition 2.3.28. Let C be a class of graphs and let a > 0. Then
C∇a :=⋃G∈C
G∇a.
Definition 2.3.29. Let C be a class of graphs. Then
`dens(C∇) := supi→∞
lim supG∈C∇i
log ||G||log |G|
.
Note that if none of the graphs in an infinite class of graphs C contain an edge,`dens(C∇) = −∞. If we exclude this (rather uninteresting) case, we obtain the followingtrichotomy:
Theorem 2.3.30 (Class Trichotomy, Nesetril and Ossona de Mendez [133]). Let C be an
infinite class of graphs such that at least one graph in C contains an edge. Then `dens(C∇)can only take the values 0, 1 or 2. Moreover
• C is of bounded size (or asymptotically edgeless) if, and only if, `dens(C∇) = 0;
• C is nowhere dense if, and only if, `dens(C∇) ∈ 0, 1;
Remark. In the third case, i.e. when `dens(C∇) = 2, we say that C is somewhere dense.
This also allows us to define a special case where the ratio ||G||/|G| is bounded bya constant c(t) for every G ∈ C∇t, calling these kinds of classes, classes with boundedexpansion.
Definition 2.3.31. Let C be a class of graphs. Then C has bounded expansion if for everyt > 0 there is c(t) such that ||G|||G| < c(t) for every G ∈ C∇t.
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There is an interesting connection between these classes and the parameter tree-depth(cf. Definition 4.0.4) introduced below. Both nowhere-dense classes and bounded expan-sion classes can be characterised using colourings where each colour induces a graph withlow tree-depth.
Definition 2.3.32. Let G be a graph and p ∈ N. Then we denote by χp(G) the leastnumber of colours required to properly colour G in such a way that every set of coloursS with |S| ≤ p induces a subgraph of H with tree-depth at most |S|. We call such acolouring a low tree-depth colouring.
Theorem 2.3.33 (Nesetril and Ossona de Mendez [131, 133]). Let C be a class of graphs.Then
• C has bounded expansion if, and only if, for every integer p there is an integer Np
such that for any graph G ∈ C we have χp(G) ≤ Np.
• C is nowhere dense if, and only if, for every ε > 0 we can find sufficiently large p,Nε
such that for all G ∈ C with |G| > Nε we have χp(G) ≤ |G|ε.
2.4 Order theory
When we talk about graphs, it is often easier to think about orders on the vertices insteadof defining additional structures. In this section we introduce some elementary notions oforder theory, leading to the new notion of an elimination order. This notion is used toestablish an alternative characterisation of tree-depth in Chapter 4, and is crucial for thestudy of elimination distance in Chapter 5.
A pre-order captures important aspects of the intuitive idea of an ordering of a set.
Definition 2.4.1. A pre-order (or quasiorder) is a binary relation ≤ on a set S which isreflexive and transitive.
Remark. For two elements a, b in a set S partially ordered by ≤, we write a < b as anabbreviation for a 6= b and a ≤ b.
The notion of partial order is a formalisation of the intuitive idea of an ordering of aset. It is like a pre-order, but is also antisymmetric.
Definition 2.4.2. A partial order is a binary relation ≤ on a set S which is reflexive,antisymmetric and transitive.
In a partial order there is no requirement that any two elements have to be comparable.If we require this, an order is called total :
Definition 2.4.3. A partial order ≤ on S is a total order (or linear order) if either a ≤ bor b ≤ a for all a, b ∈ S.
If all the elements of a subset of a partially ordered set are comparable, it is called achain.
Definition 2.4.4. Let S be a set and let ≤ be a partial order on S. We say a set T ⊆ Sis a chain if it is totally ordered by ≤.
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This allows us to introduce a notion of height of an order.
Definition 2.4.5. Let S be a set and let ≤ be a partial order on S. We say that ≤ hasheight k if the length of the longest chain in it is k.
Sometimes we want to restrict the order relation of a partially ordered set, so thatit only holds between comparable elements that are immediate neighbours (i.e. distinctwithout an intermediate element). For example, in the case of a linear order, we mightonly be interested in the successor relation. This is called a covering relation.
Definition 2.4.6. Let S be a set and let ≤ be a partial order on S. For two elementsa, b ∈ S we say b covers a if a < b and there is no c ∈ S such that a < c < b. This relationis called the covering relation of ≤.
Note that if we require that for each element the predecessors form a chain, we obtaina partial order with a covering relation that is a directed tree. This is because any twopredecessors must be comparable, so there can only be one path that contains both ofthem.
Definition 2.4.7. If ≤ is a partial order on S, and for each element a ∈ S, the setb ∈ S | b ≤ a is a chain, we say ≤ is a tree order.
Remark. Note that the covering relation of a tree order is not necessarily a tree, but maybe a forest.
Analogue to trees we define the level of a vertex in a tree order.
Definition 2.4.8. Let G be a graph and let ≤ be a tree order for G. The level of a vertexv ∈ V (G) is the length of the chain w ∈ V (G) | w ≤ v. We denote the level of v bylevel<(v).
If we consider the tree-orderings of the vertices of a graph that additionally requirethat adjacent vertices are comparable we obtain a notion similar to that of a tree-depthdecomposition (defined below in Definition 4.2.1).
Definition 2.4.9. Let G be a graph. An elimination order ≤ of G is a tree order on thevertices of G, such that for each edge uv ∈ E(G) we have either u ≤ v or v ≤ u.
Remark. We show below in Proposition 4.2.3 that this gives us an order-theoretic char-acterisation of tree-depth (introduced in Chapter 4): there is an elimination order ≤ ofheight k for a graph G if, and only if, td(G) ≤ k.
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CHAPTER 3
Distance parameters for graphproblems
This chapter introduces several graph parameters. Graph parameters are numeric prop-erties of graphs that are used in the study of the parameterized complexity of graphproblems.
The graph parameters introduced in this section are all of a certain kind. We say thata graph parameter is a distance parameter if it measures how far a graph is from beingcontained in some class of graphs. This is one of the central notions of this dissertationand all of the parameters discussed in this section are distance parameters. Figure 1.1demonstrates the relationships between the distance parameters introduced here (exceptfor clique-width).
That a graph parameter is a measure of distance is not always immediately apparentfrom its definition. It is obvious for a parameter such as deletion distance to a class C,as it is just the number of vertices we need to remove to place a graph in a certain classC, but also applies to parameters such as tree-width, which measures how far a graph isfrom being a tree. We make clear for each parameter discussed in this chapter why weconsider it to be a notion of distance.
The distance parameters we are concerned with in this dissertation measure distancesto classes of sparse graphs. We do not define sparsity precisely, but intuitively it meansthat the graphs in such classes do not contain many edges. For example the class of trees isa a sparse graph class, and the notion of nowhere dense graph classes (see Definition 2.3.24)defines a very general notion of sparse graph classes. Note that sparsity is a property ofgraph classes, not single graphs, and a graph class can still be sparse even if it containssome dense graphs (e.g. complete graphs). We will use the terms sparse graph class andclass of sparse graphs interchangeably.
Distance parameters are widely studied by researchers in parameterized complexitytheory. The main goal of that research is to study different parameterizations of a problemto gain a better understanding of the problem’s hardness through a refined analysis ofthe different parameters, and subsequently identify parameterizations that lead to fixed-parameter tractability. Recall from our discussion in the introduction that there aregenerally two motivations for studying a parameter in parameterized complexity: theparameter itself can be the focus of interest, or the parameter is mainly a tool to betterunderstand a problem. The study of distance parameters allows both: Consider as an
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example the problem Vertex Cover: given a graph G and an integer k, the question iswhether G has a vertex cover (defined below in Definition 3.2.1) of size at most k. We seebelow that the vertex cover number, i.e. the size of a minimum vertex cover in a graph Gis just the vertex-deletion distance of G to the class of edge-less graphs.
Distance parameters pose an interesting problem by themselves: determining the dis-tance of an input graph G to a class C is, in general, a computationally challenging problemin its own right. For example, the Vertex Cover problem is NP-hard and often studiedwith the vertex cover number of the graph as parameter.
On the other hand, the parameter can be used as a way to better understand whatkind of restrictions make other problems tractable. A canonical example is again VertexCover: a popular parameterization in the study of many problems is the vertex covernumber of a graph.
Distance parameters suggest a certain strategy to obtain fixed-parameter tractabilityresults. If a problem one wishes to solve is tractable on the class C, then distances toC provide interesting parameterizations of that problem. Guo et al. [95] recognized thisas a pattern: when studying the complexity of a parameterized problem, often fixed-parameter tractability results are obtained by considering a class of graphs C where theproblem is tractable, i.e. there is a known polynomial-time algorithm that solves theproblem restricted to C, and parameterizing the problem by some notion of distance tothat class C. They coined the term ‘distance to triviality’ for this and suggested it as ageneral strategy, giving several examples to demonstrate its fruitfulness.
We only discuss parameters in this chapter that are used in one or more places through-out the dissertation. There are many other interesting distance parameters, such as thecrossing number of a graph, which is a notion of distance to the class of planar graphs,to give just one example of a distance parameter not mentioned.
Note that we defer the introduction of the distance parameter tree-depth to Chapter 4,which is dedicated to it. Tree-depth receives special attention in this dissertation, becauseof its relationship to Chapter 5, where we introduce the new graph parameter eliminationdistance to a class C. Elimination distance naturally generalises the notion of deletiondistance, with tree-depth being a special case. Tree-depth also has importance in Chap-ter 8, where we apply the method developed in that chapter to establish that tree-depthis FPT.
3.1 Deletion distance to a class CA popular template for graph parameters is the deletion distance to some class of graphsC.
Definition 3.1.1. A graph G has deletion distance k to a class C if there are k verticesv1, . . . , vk ∈ V (G) such that G \ v1, . . . , vk ∈ C.
The study of the parameterized complexity of deletion distances was motivated by apowerful hardness result: Lewis and Yannakakis [112] show that vertex deletion to a classC is NP-hard on graph classes C that are non-trivial and closed under induced subgraphs.This means that it is unlikely that we will discover efficient algorithms that check whethera graph has a certain distance on the class of all graphs.
The search for fixed-parameter tractability results for deletion distances was initiatedby Cai [32], and all kinds of vertex deletion problems are now being studied in that
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context. Deletion distances have many applications and are now among the most studiedproblems in the world of parameterized complexity.
A large number of parameterized parameters are either directly deletion distance prob-lems, or can be rephrased in that way. For example Cluster Vertex Deletion (dele-tion distance to the class of disjoint unions of cliques), Graph Bipartization (deletiondistance to the class of bipartite graphs), and Feedback Vertex Set (deletion distanceto the class of forests) are all problems where the natural parameterization (by the outputsize) is a distance to a graph class.
Deletion distance is used in Section 8.3.1 (where we show that if C is nowhere denseand first-order definable, then the problem of determining the deletion distance to C isslicewise first-order definable and slicewise nowhere dense and thus FPT), and Section 6.2(where we show that Graph Isomorphism is FPT parameterized by the deletion distanceto bounded degree graphs).
3.2 Vertex cover number
A vertex cover of a graph G is a set of vertices A ⊆ V (G) so that each edge of G is incidentto at least one vertex in A. As mentioned above, the problem Vertex Cover is theproblem of deciding whether there is a vertex cover of a given graph of at most a givensize. Determining the vertex cover number of a graph is a classical optimization problem,and deciding whether there is a vertex cover of at most a given size is NP-complete – infact, it is on the list of Karp’s [103] original NP-complete problems.
Definition 3.2.1. Let G be a graph. A vertex cover of G is a set of vertices S ⊆ V (G)such that for all edges uv ∈ E(G) either u ∈ S or v ∈ S.
The size of a minimal vertex cover is a popular parameter called the vertex covernumber.
Definition 3.2.2. Let G be a graph. The vertex cover number of G, written vc(G), isthe size of a minimal vertex cover of G.
Remark. Note that the vertex cover number is in fact a deletion distance – it is just thedeletion distance to the class of edgeless graphs.
The vertex cover number is possibly the most popular parameter studied in param-eterized complexity theory and there has been a long sequence of papers proving theexistence of smaller and smaller kernelizations for Vertex Cover.
3.3 Edit distance to a class CInstead of just removing vertices from a graph, we can also consider more general editingoperations on a graph; e.g. removing or adding an edge.
In the following we abbreviate the editing operations vertex deletion, edge deletion,vertex addition and edge addition as dv, de, av and ae respectively. Then for each non-empty O ⊆ dv, de, av, ae we define the O-edit-distance as follows:
Definition 3.3.1. A graph G has O-edit-distance k to a class C if we can obtain a graphG′ ∈ C from G using k editing operations from O.
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Many graph editing problems are hard in general. In addition to the result by Lewisand Yannakakis [112], discussed above in Section 3.1 who show that many graph deletionproblems are NP-hard, Yannakakis [159] also shows that this holds for edge deletionproblems on many graph classes. This hardness motivates the study of the parameterizedcomplexity of graph editing problems.
Just as for deletion distance, many problems of this form are studied in parameter-ized complexity. For example Cluster Editing (edit distance to the class of disjointunions of cliques), Twin Graph Editing (edit distance to the class of twin graphs) andChordal Editing (edit distance to the class of chordal graphs) are generalisations ofthe versions mentioned above in Section 3.1.
In Section 8.3.2 we demonstrate how our result from Chapter 8 can be applied whenC is a class of (sparse) graphs defined by degree constraints.
3.4 Path-width & tree-width
Tree-width is one of the most studied graph parameters in parameterized complexity.The concept was initially introduced under the name of dimension by by Bertel andBrioschi [15]. After being rediscovered by Halin [97] it was again rediscovered and popu-larized by Robertson and Seymour [141] in the context of graph minor theory.
At the core of the definition is a notion of a tree decomposition of a graph, whichmakes it fairly easy to apply dynamic programming techniques. Path decompositions area special case of tree decompositions, where the decomposition is a not just a tree, but apath.
Definition 3.4.1. Let G be a graph. A tree decomposition of G is a pair (T,B) where Tis a tree with nodes B, called the bags of the tree decomposition. Each node in B is a setof vertices of G that satisfy the following:
• V (G) =⋃B∈B B;
• for each uv ∈ E(G) there is a B ∈ B such that u, v ∈ B;
• for each v ∈ V (G), the subtree induced by B ∈ B | v ∈ B is connected.
If T is a path, then we call (T,B) a path decomposition.The width of a path or tree decomposition is one less than the size of the largest bag,
i.e.
maxB∈B|B| − 1.
Definition 3.4.2. The tree-width of a graph G, abbreviated as tw(G), is the minimalwidth of a tree decomposition of G.
Definition 3.4.3. The path-width of a graph G, abbreviated as pw(G), is the minimalwidth of a path decomposition of G.
Remark. Since every path decomposition is a tree decomposition, we have that the tree-width of a graph is bounded above by its path-width.
Tree-width (and path-width) are lower bounds for vertex cover number.
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Proposition 3.4.4. Let G be a graph with vertex cover number k. Then G has path-widthand tree-width at most k.
Proof. Let A be a vertex cover of size k of G. For each v ∈ V (G) \ A define the set
Bv := A ∪ v.
Let B := Bv | v ∈ V (G) \ A. Let T be a path with node set B, in any order. Then(T,B) is a path decomposition of G (and thus also a tree decomposition).
Note that all the bags have size k+ 1, thus the path-width (and thus also tree-width)of G is at most k.
We study a problem (the Anchored k-core problem) parameterized by tree-widthin Chapter 9.
3.5 Clique-width
Clique-width was introduced as a graph parameter by Courcelle and Olariu [43] to measurethe complexity of graphs in terms of hierarchical decompositions. The clique-width of agraph is defined as the smallest amount of colours needed to construct the graph with acertain set of rules.
To formally define clique-width, we first need to introduce some notation.
Definition 3.5.1. Let k > 0. A graph G is k-constructible if there is a colouringχ : V (G) → 1, . . . , k of G such that (G,χ) can be constructed using the followingoperations:
• introduce: Create a new graph H with a single vertex v and colouring χH such thatχH(v) = i for some 1 ≤ i ≤ k. The one-vertex graph with vertex v and colour i isdenoted i(v).
• disjoint union: Take the disjoint union of two graphs.
• join: Add an edge between every vertex of colour i and every vertex of colour j forsome 1 ≤ i, j ≤ k.
• recolour: Colour all vertices of i with colour j for some 1 ≤ i, j ≤ k.
Remark. These operations can be defined in algebraic terms, so that there is an algebraicexpression that constructs a k-constructible graph (with a colouring).
Definition 3.5.2. Let G be a graph. The clique-width of G is the smallest integer k suchthat G is k-constructible.
Clique-width is a distance parameter in the sense that it measures the number ofcolours needed to construct a graph from the class containing only the empty graph.
Although it is certainly a distance to a (very) sparse graph class in this sense, graphclasses with bounded clique-width can contain dense graphs as well, e.g. it is easy to seethat every complete graph has clique-width 1. So clique-width does not give rise to sparsegraph classes, i.e. the class of graphs with clique-width at most k is not a sparse graphclass for k > 0 and for example contains all cliques. However, Gurski and Wanke [96] showthat clique-width can in a certain way be thought of as a generalisation of tree-width; aclass of graphs of bounded clique-width that is also sparse must have bounded tree-width:
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Theorem 3.5.3 ([96]). Let G be a graph with clique-width k, and let n > 1. If G doesnot contain the complete bipartite graph on n vertices Kn,n as a subgraph, then G hastree-width at most 3k(n− 1)− 1.
Moreover, Courcelle [40] showed the following:
Theorem 3.5.4 (Courcelle’s Theorem). Every problem that can be expressed in monadicsecond-order logic has a linear time algorithm on graphs of bounded tree-width.
Later the theorem was generalised to clique-width by Courcelle et al. [42]:
Theorem 3.5.5. Every problem that can be expressed in monadic second-order logic hasa linear time algorithm on graphs of bounded clique-width.
As the clique width of a graph is bounded in terms of its rank width, the decompositioncan be found via an algorithm for rank-width.
We use this generalisation of Courcelle’s Theorem in Chapter 9, where we study aproblem (the Anchored k-core problem) parameterized by clique-width.
3.6 Max leaf number
The max leaf number of a connected graph G is just the maximum number of leaves in aspanning tree for G. (We give the relevant definitions below.) Fellows et al. [66, 68] haveinitiated its study as a parameter in the context of parameterized complexity.
The max leaf number of a graph is formally defined as follows:
Definition 3.6.1. Let G be a connected graph. A spanning tree of G is a connectedsubgraph of G that is a tree.
Definition 3.6.2. The max leaf number of a connected graph G is the largest number ofleaves in any of its spanning trees.
We can subdivide an edge uv in a graph G by removing the edge uv, adding a newvertex w and adding the edges uw and wv. A graph G′ that is obtained from a graph Gby subdividing edges is called a subdivision of G. Kleitman and West [106] show that ifa graph G has max leaf number k, then it is a subdivision of a graph with at most 4k− 2vertices, in particular all vertices of G, except for at most 4k−2, must have degree 2. Themax leaf number therefore gives us a deletion distance to the (very sparse) graph class ofdegree-2 graphs, that is, paths and cycles. It is thus a distance to a sparse graph class.
The above also demonstrates that if a graph has max leaf number k it can have tree-width at most 4k−3, because subdividing edges does not affect the tree-width of a graph.So both bounded vertex cover number and bounded max leaf number imply boundedtree-width. However, neither of them implies a bound on the other parameter: A stargraph has a vertex cover of size 1, while it can have unbounded max leaf number. A pathhas max leaf number 2, while it can have unbounded vertex cover number.
We introduce the max leaf number of a graph in this section because the parametergeneralised tree-depth, discussed in Section 4.4, is explicitly designed to be a commongeneralisation (i.e. a parameter that gives a lower bound) for both max leaf number andtree-depth.
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3.7 Degeneracy and the k-core of a graph
Degeneracy is yet another measure for how sparse a graph is. It was first introducedunder the name colouring number by Erdos and Hajnal [59] as the least number k forwhich there is an ordering of the vertices such that every vertex has at most k neighboursthat come before it in the ordering. Lick and White [114] gave an equivalent definitionunder the name degeneracy. They defined a graph to be k-degenerate for some integer kif every subgraph contains a vertex with at most k neighbours. This is the more commondefinition and the one we will use.
Definition 3.7.1. Let k ≥ 0. A graph G is k-degenerate if every subgraph of G containsa vertex of degree at most k. We say that the least k for which G is k-degenerate is thedegeneracy of G.
The degeneracy of a graph can be interpreted as a distance parameter in the followingway: since every subgraph of a graph G with degeneracy k contains a vertex of degree atmost k, we can, starting with the whole graph G, keep isolating vertices by deleting atmost k edges from the graph. Thus we can isolate all the vertices with at most k|G| edgedeletions, and the graph has thus an edge deletion distance (or de-edit-distance) of atmost k|G| to the class of edgeless graphs.
Bodlaender et al. [19] show that the degeneracy of a graph is a lower bound for itstree-width:
Lemma 3.7.2 (Bodlaender et al. [19], Lemma 3). Let G be a graph. Then the degeneracyof G is at most the tree-width of G.
The degeneracy of a graph is also called its k-core number for the following reason:The k-core of a graph G is what is left of a graph when we, repeatedly, remove all verticesfrom G that have less than k neighbours; or formally:
Definition 3.7.3. Let G be a graph and k ≥ 0. The k-core of G is a maximal subgraphof G with minimum degree k. The largest k for which the k-core of G is non-empty iscalled the k-core number of G.
Remark. Note that there is one unique k-core. We obtain the k-core by repeatedly re-moving all vertices of degree less than k.
The k-core is connected to k-degeneracy in the following way: If the k-core is non-empty, then G must have degeneracy at least k. Conversely the degeneracy of G is thelargest k for which G has a non-empty k-core, i.e. the k-core number of G.
The degeneracy (and k-core) of a graph will be used in Chapter 9 in the study of theAnchored k-core problem. A class of graphs of bounded degeneracy is itself a sparsegraph class, and we will consider a notion of distance to such classes in Chapter 9.
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CHAPTER 4
Tree-depth: Exploring the spacebetween vertex cover number andtree-width
This chapter has the dual purpose of showing why tree-depth is an interesting parameter,as well as being a preparation for the next chapter. In Chapter 5 we introduce the newparameter elimination distance to a class C, which is a generalisation of tree-depth.
The tree-depth of a graph G is (non-recursively defined to be) the minimum height ofa directed forest that contains G in its symmetric transitive closure. Formally, tree-depthis recursively defined as follows:
Definition 4.0.4 (Nesetril and Ossona de Mendez [130]). Let G be a graph. We writetd(G) for the tree-depth of G, which is defined as follows:
td(G) :=
0, if V (G) = ∅;1 + mintd(G \ v) | v ∈ V (G), if G is connected;
maxtd(H) | H a component of G, otherwise.
This chapter is organised as follows. After first giving some background in Section 4.1,we establish a number of equivalent characterisations in Section 4.2. Tree-depth is a veryversatile parameter: the combinatorial definition in terms of a tree-depth decomposition(see Definition 4.2.1) makes it an ideal target for dynamic programming approaches. Butit can also be defined recursively (see Definition 4.0.4) or even order-theoretically (seeDefinition 2.4.9). In Section 4.3 we show that tree-depth is a parameter that is locatedin-between the (more popular) parameters vertex cover number and tree-width. We thendiscuss the parameter generalised tree-depth in Section 4.4, a parameter introduced byBouland et al. [23] that is based on a variant of a game that defines tree-depth. Thegeneralised tree-depth of a graph is a lower bound for both its tree-depth and its max leafnumber. Next, in Section 4.5 we prove fixed-parameter tractability and hardness resultsfor some problems parameterized by tree-depth:
• Precolouring Extension: We show that Precolouring Extension is W[1]-hard parameterized by tree-depth. We give a reduction from the MulticolourClique problem to List Colouring, and from List Colouring to Precolour-ing Extension.
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• Boxicity: We state an additive 1-approximation algorithm for Boxicity param-eterized by tree-depth. We show that, given a graph G, we can find a (b + 1)-boxrepresentation of G in linear FPT time such that the boxicity of G is either b orb+ 1.
Lastly we discuss some open questions related to tree-depth in Section 4.6.
4.1 Background
Nesetril and Ossona de Mendez [130] introduced the name tree-depth and established itsclose connection to nowhere-dense and bounded expansion graph classes: It is possible tocharacterise nowhere-dense and bounded expansion graph classes in terms of a partitioninto graphs of bounded tree-depth (cf. Definition 2.3.32 and Theorem 2.3.33). However,even though this revived the interest in the parameter, tree-depth was not a new concept– it was studied before under the names of vertex ranking number ordered chromaticnumber, minimum elimination tree height and rank function [18, 54, 134, 145].
Tree-depth is also closely related to the parameter elimination distance to a class Cthat we introduce in Chapter 5 as a new distance parameter. We show in Section 5.4that tree-depth is in fact the special case of elimination distance to the class of edgelessgraphs. The notion of deletion in tree-depth is that of elimination, which allows forparallel deletion in each component at each deletion step. So tree-depth is also clearly adistance parameter.
A useful fact about graphs of bounded tree-depth is that they do not contain longpaths. We give a proof of this here.1
Lemma 4.1.1. Let k ≥ 1 and let G be a graph with tree-depth at most k. Then G doesnot contain a path of length greater than 2k − 2.
Proof. The proof is by induction on k. We assume without loss of generality that G isconnected, because otherwise we can just prove the statement for each component. Ifk = 1, then G only contains isolated vertices and no paths.
Suppose k > 1 and no graph with tree-depth less than k contains a path of lengthgreater than 2k−1−2. Let v be a vertex so that all components of G\v have tree-depthless than k. Then none of these components contain a path of length greater than 2k−1−2.A path of length greater than 2k−1 − 2 in G must therefore contain v and is of length atmost (2k−1 − 2) + (2k−1 − 2) + 2 = 2k − 2.
4.2 Alternative characterisations of tree-depth
In the following we give a number of different characterisations of tree-depth. It is a veryversatile parameter and the different definitions are useful in different situations.
The first characterisation is in terms of a decomposition of the graph. We now showthat the definition given above (Definition 4.0.4) is equivalent to the following definitionin terms of a tree-depth decomposition:
Definition 4.2.1. Let G be a graph. A tree-depth decomposition of G is a rooted directedforest that contains G in its symmetric transitive closure.
1This fact also follows from the discussion in Section 6.2 in [133], but we give a direct proof here.
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Proposition 4.2.2. Let G be a graph. The tree-depth of G is the minimum height of atree-depth decomposition of G.
Proof. Assume without loss of generality that G is connected. We prove the statementby induction. Let k = 0. Then the graph has tree-depth 0 and an empty, i.e. height 0,tree-depth decomposition.
Let k > 0 and suppose the statement holds for smaller values. Suppose G has tree-depth k. Then there is v ∈ V (G) such that G\v has tree-depth k−1. By the inductionassumption there is a tree-depth decomposition of G\v of height k−1. We can constructa tree-depth decomposition of G by adding v as a root to that tree-depth decomposition,which then has height k.
Conversely assume that there is a tree-depth decomposition of height k of G. SinceG is connected, it has a single root v. If we remove the root v, we obtain a forest oftree-depth decompositions, each for a component of G \ v, and each of height at mostk − 1 (but at least one of height exactly k − 1). Thus, by the induction assumption,G \ v has tree-depth k − 1, and G has tree-depth k.
There is also an order theoretic characterisation of tree-depth, using the notion of anelimination order defined in Definition 2.4.9. Instead of considering the tree in Defini-tion 4.2.1 as a separate graph defined on the same vertices, we can view it as a tree-orderdefined on the vertices.
Proposition 4.2.3. A graph G has td(G) ≤ k if, and only if, there is an eliminationorder of height at most k for G.
Proof. We assume without loss of generality that G is connected and prove this by induc-tion on k.
If k = 0, then G must be empty (by both definitions) and the statement holds.Suppose k > 0 and the statement holds for smaller values. First assume that G has
tree-depth k. Then there is v ∈ V (G) such that G \ v has tree-depth k − 1. By theinduction assumption there is an elimination order of G\v where the longest chain haslength k − 1. We obtain an elimination order for G by adding v as the minimal elementto that order. The longest chain has now length k, so it has height k.
Conversely assume that there is an elimination order ≤ of height k for G. Note thatsince G is connected, there is a unique minimal element: Suppose for a contradiction thatthere are two minimal elements v, v′. Since G is connected, there is a path P connectingv and v′. In the elimination order ≤ every pair of adjacent vertices must be comparable,so every neighbouring pair of vertices on P must be comparable. Since v and v′ areboth minimal, there must be vertices a, b, c on P such that b < a, b < c and b, c areincomparable. But this is impossible in a tree order.
Let v be the single minimal element of G. If we remove v, we obtain an eliminationorder of height k−1 for G\v. Thus, by the induction assumption, G\v has tree-depthk − 1, and G has tree-depth k.
Another characterisation of tree-depth comes from games [79, 85]. Consider the fol-lowing game played on a connected graph G. The game has two players: Cop player andRobber player. The Robber player controls one robber that occupies a vertex of G andthat she can move along edges of G in each round. The Cop player tries to catch therobber, choosing a vertex of G in each round where he places a cop. The Cop playercannot move the cops after placing them.
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At the beginning of the game the Robber player places the robber on some vertex inV (G). Then the Cop player announces where he will place his first cop, which can beanywhere in the graph including the current position of the robber.
After i−1 rounds of the game, there are i−1 cops placed on vertices of the graph andthe robber is placed on some vertex v. The Robber player can move the robber aroundalong edges in the graph, but the robber cannot be moved through positions occupied bythe first i−1 cops. If the robber shares a vertex with a cop at the beginning of the round,it can be moved away (but only if the neighbours are not also occupied by cops). Thenthe Cop player places a new cop on the graph.
The Cop player wins the k-round game if he can catch the robber in at most k rounds,i.e. if the robber shares a vertex with a cop at the end of a move and the Robber playerwould not be able to move the robber away in the next round. The Robber player winsthe k-round game if after k rounds all cops are placed somewhere on the graph and therobber is still not caught.
This game characterises tree-depth in the following way [23]; the tree-depth of k isexactly the least number of rounds needed for the Cop player to have a winning strategyin the game.
Proposition 4.2.4. Let G be a graph and let k > 0. The tree-depth of G is less than kif, and only if, the Robber player has a winning strategy in the k-round game.
4.3 Relation to vertex cover number and tree-width
In this section we show that tree-depth as a distance parameter is located in-between (cf.Figure 1.1) the (more popular) parameters vertex cover number (cf. Definition 3.2.2) andtree-width (cf. Definition 3.4.2) in the sense that for any graph G we have
tw(G) < td(G) ≤ vc(G) + 1.
We give a proof of both these statements in the following:
Proposition 4.3.1. Let G be a graph with vertex cover number k. Then G has tree-depthat most k + 1.
Proof. Let A be a vertex cover of size k of the graph G. We use the alternative character-isation of tree-depth given in Definition 4.2.1 and observe that the k vertices of the vertexcover A, arranged as a path in any order, with the remaining vertices at the bottom, givea tree-depth decomposition of the graph of height k + 1.
Next we show that tree-depth is an upper bound for tree-width.
Proposition 4.3.2. Let G be a (non-empty) graph which has tree-depth k. Then G hastree-width at most k − 1.
Proof. Let G be a graph with tree-depth k. We prove the statement by induction.Suppose G only has a single vertex v. Then the only bag is B := v and this gives
us a one-node tree decomposition, which has width 0. This can be easily generalized to agraph with tree-depth 1, i.e. a graph that only contains isolated vertices.
Otherwise assume k > 1 and that the statement holds for graphs of tree-depth lessthan k. Since k > 1, there is a vertex v ∈ G such that G \ v has tree-depth k − 1. Bythe induction assumption there is a tree decomposition of G \ v of width at most k− 2.Adding v to every bag gives us a tree decomposition of G of width at most k − 1.
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Remark. Using the alternative characterisation of tree-depth in Definition 4.2.1 we canexplicitly construct the tree decomposition by making each branch of the tree-depth de-composition (that is, the vertices on a path from the root to a leave of the tree-depthdecomposition) a bag of the tree decomposition. This actually gives a path decomposition,and shows that tree-depth is in fact in-between vertex cover number and path-width.
This puts tree-depth in an interesting spot: Vertex cover and tree-width are very pop-ular parameters and a large number of problems have been studied parameterized by oneor both of them. Many of them have been shown to be fixed-parameter tractable param-eterized by vertex cover number, and on the other hand W[1]-hardness parameterized bytree-width (or path-width) is either conjectured or proven. Determining the complexitystatus of such problems with regard to tree-depth would help to determine where exactlythe line between tractability and intractability is for them. This motivates our study ofseveral problems in Section 4.5, and we discuss some further problems in Section 4.6.
4.4 Generalised tree-depth
Bouland et al. [23] introduce a new parameter called generalised tree-depth with the aim ofgeneralising both tree-depth and max leaf number, i.e. so that the generalised tree-depthof a graph G gives a lower bound of both the tree-depth of G and the max leaf numberof G.
It is defined via a modified version of the Cop and Robber game described above inSection 4.2. Now the Cop player wins if the robber is confined to either a simple pathwith cops at both endpoints or a simple cycle. The endpoints of this path (which must beoccupied by cops) may be connected to other vertices of the graph, but the other verticesin the path may not be adjacent to any vertices in the rest of the graph. For the Copplayer to win the game by confining the robber to a cycle, the cycle must be disconnected(by cops) from the rest of the graph.
Definition 4.4.1. Let G be a graph. Then the generalised tree-depth of G is the leastk for which the Cop player has a winning strategy in the modified k-round Cops andRobber game described above. We write gtd(G) for the generalised tree-depth of a graphG.
In Chapter 5 we introduce the parameter elimination distance to a class C, and wealready mentioned that tree-depth is a special case. We see in Section 5.4.1 that gener-alised tree-depth gives us another special case: elimination distance to the class C, whereC is a subclass of the graphs with degree at most 2.
4.5 Applications
Now we look at problems parameterized by tree-depth. In this section we consider theproblems Precolouring Extension and Boxicity parameterized by tree-depth. Weshow that, with this parameter, Precolouring Extension is W[1]-hard and that thereis an additive 1-approximation algorithm for Boxicity that runs in linear FPT time.
Recall from Section 4.3 that tree-depth as a parameter is ‘sandwiched’ in-betweenvertex cover number and tree-width (and path-width) in the following sense:
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tw(G) ≤ pw(G) < td(G) ≤ vc(G) + 1.
As we have mentioned above in Section 4.3, there are several problems that are knownto be hard (e.g. W[1]-hard) parameterized by tree-width or path-width, and known tobe fixed-parameter tractable parameterized by vertex cover number. Examples of thisinclude Boxicity [4], Cutwidth [45], as well as several colouring problems [69]. So theborder between tractability and intractability for these problems must lie somewhere inbetween vertex cover number and tree-width. Considering a parameterization by tree-depth allows us to zero in on the line between fixed-parameter tractability and hardness.In the following we exhibit results for two such problems: Precolouring Extensionand Boxicity.
4.5.1 Hardness: List Colouring and Precolouring Extension
In the following we show that Precolouring Extension is W[1]-hard parameterizedby tree-depth.
The problem is known to be fixed-parameter tractable parameterized by vertex covernumber (shown by Fiala et al. [69]), and W[1]-hard parameterized by tree-width (shown byFellows et al. [63]). So a natural question is: is it fixed-parameter tractable or W[1]-hardparameterized by tree-depth? We show that it is in fact W[1]-hard.
The problem is defined as follows:
Pre-Colouring ExtensionInput: A graph G, a list of colours L, and a colouring cW : W → L for a set ofvertices W ⊆ V (G)Parameter: td(G)Problem: Is there a proper colouring c : V (G) → L for G such that c(v) = cW (v)for all v ∈ W?
The proof we give in this section uses a reduction from the List Colouring problem,which is defined as follows:
List ColouringInput: A graph G, a list of colours L, and an assignment ` : V (G)→ 2L of allowedcolours for each vertexParameter: Vertex cover number of GProblem: Is there a proper colouring c : V (G) → L such that c(v) ∈ `(v) for allv ∈ V (G)?
The result List Colouring is W[1]-hard parameterized by vertex cover number isknown2, but we are not aware of a proof in the literature, so, for the sake of completeness,we give one here.
The proof we give here is a reduction from Multicolour Clique, which was shownto be W[1]-hard by Fellows et al. [65] through a reduction from Clique.
The Multicolour Clique is defined as follows:
2It is mentioned in [67], but no proof or reference is given.
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Multicolour CliqueInput: An integer k, and a properly k-coloured graph GParameter: kProblem: Does G contain a k-clique consisting of one vertex of each colour?
Theorem 4.5.1. List Colouring is W[1]-hard parameterized by vertex cover number.
Proof. We give a reduction from the Multicolour Clique problem. Our argumentmimics and extends the argument given by Fellows et al. [63, Theorem 1] to show thatList Colouring is W[1]-hard parameterized by tree-width.
Let G be the given graph, and let c be the given proper colouring of G. We constructa graph G′ and lists of allowed colours as follows. Let the global list of colours we use bethe vertex set of G, i.e. define L := V (G). The vertices V (G′) are as follows:
• For each 1 ≤ i ≤ k we have a vertex vi. We define `(vi) := c−1(i), i.e. the availablecolours in G′ are the vertices of colour i in G;
• For each non-edge uw 6∈ E(G) with c(u) 6= c(w), we add a vertex vuw adjacent tovc(u) and vc(w). The list of available colours for vuw is `(vuw) := u,w.
Note that G′ is a bipartite graph, with v1, . . . , vk being one of the sides. Sov1, . . . , vk is a vertex cover for G′, and the vertex cover number of G′ is at most k.
Suppose G has a multicolour clique. Then we can colour G′ as follows: Assign to eachvi the vertex from the clique as the colour. All the vertices vuw correspond to non-edges,so it cannot be that both u and w are part of the clique and one of them must be freeand available as a colour for vuw.
Conversely assume there is a valid list colouring c for G′. We argue that the verticesc(v1), . . . , c(vk) assigned as colours to v1, . . . , vk form a clique in G.
We have to show that between any pair of these vertices there is an edge. Considertwo such vertices vi and vj with 1 ≤ i < j ≤ k. First note that c(vi) (with colour i in G)has a different colour from c(vj) (with colour j in G) because i 6= j. Now observe thatbecause they are in different colour classes, by our construction there is no edge betweenc(vi) and c(vj) if, and only if, they appear on the list of a vertex vc(vi)c(vj) ∈ V (G′). Butthere can be no such vertex vc(vi)c(vj) because it would be adjacent to both vi and vjwith colours c(vi) and c(vj) respectively, so there would be no free colour available forvc(vi)c(vj). Thus c(vi)c(vj) ∈ E(G), and the c(v1), . . . , c(vk) form a clique. This completesthe reduction.
Remark. Note that the vertex cover v1, . . . , vk is also an independent set, so we areproving a more general statement here: the graph we are reducing to is in fact a bipartitegraph with one side bounded by the parameter.
Next we give a reduction from the List Colouring problem to PrecolouringExtension.
Theorem 4.5.2. Precolouring Extension is W[1]-hard parameterized by tree-depth.
Proof. We give a reduction from List Colouring parameterized by vertex cover number.Let G be an instance of List Colouring with allowed colours specified by the function` : V (G)→ 2L, where L = a1, . . . , a|L| is the list of all available colours.
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We obtain G′ from G as follows: For each vertex v ∈ V (G), and each colour a ∈ L,we add a vertex av, i.e. we define
W := av midv ∈ V (G), a ∈ L
and V (G′) := V (G) ∪W .We then make sure every vertex v from V (G) is adjacent to the vertices representing
the colours in L that are not allowed on its list, i.e. we define
E(G′) := E(G) ∪ vav | v ∈ V (G), a ∈ L, a 6∈ `(v).
To create a complete Precolouring Extension instance, we also define a pre-colouring cW of W as follows. We define cW to be the function that maps av to a for eachav ∈ W .
It is easy to see that our construction ensures that G has a valid list colouring if, andonly if, G′ has a proper precolouring extension.
So it is only left to show that the tree-depth of G′ is bounded by a function of k. Itis easy to see (and shown in Proposition 4.3.1) that if G has vertex cover number k, thenG has tree-depth at most k. Note that W is an independent set because we have definedno edges between vertices from W . Moreover, every vertex av ∈ W is adjacent to at mostone vertex from G, namely v. Thus a tree-depth decomposition of G′ of height k + 1 canbe obtained from a tree-depth decomposition of G by adding all the vertices from W asleaves. More precisely, we can add the vertex av ∈ W as a leaf to the branch containingv. Thus td(G′) ≤ k + 1. This completes the reduction.
4.5.2 Approximation and Parameterized complexity: Boxicity
In the following we consider the Boxicity problem. Just as Precolouring Exten-sion, it is known to be fixed-parameter tractable parameterized by the vertex covernumber of the input graph (proved by Adiga et al. [4]). However, its complexity statuswith regard to tree-width is unknown, and it is only conjectured to be hard parameterizedby tree-width [4].
For this reason, Boxicity is another natural candidate to consider for a parameter-ization by tree-depth in an attempt to get closer to an accurate understanding of wherethe problem becomes fixed-parameter tractable.
In this section we consider approximation algorithms for Boxicity. For a hard opti-mization problem, approximation algorithms and parameterized complexity are normallyconsidered to be two separate approaches to work around the hardness. However, therehas been much interest recently in combining approximation algorithms and parameter-ized complexity.
Approximation algorithms trade in some accuracy for an improved runtime. However,unlike heuristics, they provide a guaranteed runtime and a guaranteed quality of thesolution: usually the approximate solution is optimal up to a constant factor (e.g. atmost twice as large for a minimization problem). They are an increasingly popular wayto deal with NP-hard problems, see e.g. [38, 154].
Recently there has been increasing interest in combining approximation algorithmswith parameterized complexity and to study parameterized approximation algorithms,i.e. approximation algorithms that run in time O(f(k) · nc) for some constant c andcomputable function f . (See Marx [122] for an overview of some of the different approaches
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and parameterizations.) This section gives an example where combining parameterizedcomplexity with approximation algorithms has been fruitful and lead to a new result.
The specific notion we need is that of an additive p-approximation algorithm, definedas follows:
Definition 4.5.3. Let P be a graph minimisation problem that, given a graph G, asso-ciates with it an integer ξ(G). Let p > 0. An algorithm that computes a solution ξ′(G)such that ξ′(G) ≤ ξ(G) + p is called an additive p-approximation algorithm for P .
Remark. In this section we measure the runtime of the approximation algorithm solvinga parameterized problem as a function of both the parameter and the input size. Werefer to such algorithms, designed to optimize the runtime in terms of both the size of theinput and the parameter, as parameterized approximation algorithms.
In the following we exhibit a parameterized additive 1-approximation algorithm forBoxicity. The parameter we are considering is tree-depth and the algorithm runs inlinear FPT time, i.e. in time O(f(td(G) · |G|)), where G is the input graph and f is somecomputable function.
The boxicity of a graph G is the smallest dimension b that allows a b-dimensionalrepresentation of G as a box graph, i.e. it is the minimum dimension in which G canbe represented as an intersection graph of axis-parallel boxes. It was introduced byRoberts [140]. The Boxicity problem is an optimisation problem where, given a graphG, we are trying to find the boxicity of G. We do not work with box graphs here,and instead use an alternative characterisation of boxicity. In the following we define acharacterisation via intersection graphs that is more useful for our purposes here. Fora reader familiar with box graphs it should be intuitively clear that both definitions areequivalent, and we do not give a proof here.
A graph is an interval graph if it can be expressed via intervals on the real line:
Definition 4.5.4. An interval graph is a graph where each vertex can be associated withan interval on the real line so that two vertices are adjacent if, and only if, their intervalsoverlap.
We can combine multiple such graphs by considering their intersection:
Definition 4.5.5. Let G1, . . . , Gk be graphs on the same vertex set. Then a graph G onthe same vertex set is called the intersection of G1, . . . , Gk iff G contains an edge iff allthe Gi contain it.
We give a definition of boxicity using these notions:
Definition 4.5.6. The boxicity of a graph G is the smallest positive integer b such that Gis the intersection of b interval graphs. We call the b interval graphs a b-box representationof G.
In terms of this definition, the Boxicity problem is an optimisation problem wherewe are given a graph G and are trying the find the smallest integer b such that G admitsa b-box interpretation. The decision version of the problem is formally defined as follows:
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BoxicityInput: A graph G and a positive integer bParameter: td(G)Problem: Does G have boxicity at most b?
The (classical, un-parameterized, version of the) problem is NP-complete [44], and theproblem is known to be ‘resistant’ to the natural parameterization by the output value:just determining whether the boxicity of a graph is at most two is already NP-complete[107, 160]. Thus one has to look at structural parameters.
Adiga et al. [4] show that Boxicity is FPT parameterized by the vertex cover numberand also give an additive 2-approximation algorithm for Boxicity parameterized by maxleaf number (see Definition 3.6.2).
The complexity of Boxicity parameterized by tree-width is unknown, in fact Adigaet al. [4] conjecture that it is NP-hard for some graphs of constant tree-width. This makesit an interesting candidate to consider for a parameterization with tree-depth.
4.5.2.1 A parameterized approximation algorithm for Boxicity
In the following we give an parameterized additive 1-approximation algorithm for Box-icity. The parameter we are considering is tree-depth. We show that given a graph G,we can find a b-box or (b+ 1)-box representation of G in linear FPT time, where b is theboxicity of G.
The core idea of the algorithm is to compute a tree-depth decomposition of the inputgraph and reduce it to a ‘pruned tree-depth decomposition’ that only keeps a boundednumber of vertices at each level of the tree, where the ith level of a tree-depth decomposi-tion is the set of vertices with distance i to the root. We show that solving the Boxicityproblem on that smaller graph (almost) gives a solution for the whole graph. So this isessentially a kernelization argument – except that the boxicity of the kernel might be oneless in some cases.
Definition 4.5.7. Let G be a graph and let T be a tree-depth decomposition of G. Thetype with respect to T of a vertex v in G is defined as a pair (A,B) of a set of integers Aand a set of types B where
A := i | v is adjacent to a vertex at level i in T;B := type(w) | w is a child of v in T.
A pruned tree-depth decomposition of G with respect to T is a subgraph of T where ineach set of siblings we have deleted all but one copy of each type present, including thesubtrees the deleted siblings are rooted at. Formally we define it recursively as follows:
Definition 4.5.8. Let G be a graph and let T be a tree-depth decomposition of G ofheight k. If k ≤ 1, then T ′ := T is a pruned tree-depth decomposition of G with respectto T .
Otherwise, if k > 1, and R is the set of roots of T , let R′ be a minimal subset of Rwith the property that
type(r) | r ∈ R = type(r′) | r′ ∈ R′.
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Let Tr′ be a pruned tree-depth decomposition of G[V (Rr′)] with respect to the heightk − 1 tree-decomposition T [V (Rr′)], where Rr′ is the set of vertices reachable from r′ inT . Then the subgraph T ′ of T induced by the vertices
R′ ∪⋃r′∈R′
V (Tr′),
is a pruned tree-depth decomposition of G with respect to T .
Remark. Note that even though there is not a unique pruned tree-depth decomposition ofa graph with respect to some fixed tree-depth decomposition, all the pruned tree-depthdecomposition must be isomorphic.
Lemma 4.5.9. Given a graph G and a tree-depth decomposition T of G we can computea pruned tree-depth decomposition T ′ in linear time. Moreover, the size of T ′ is boundedby a function of td(G).
Proof. Let k be the tree-depth of G. We begin by computing the type of all leaves. Foreach leaf z in T , we can compute the type by looking at the at most k − 1 ancestors ofz. In each set of siblings we delete all vertices of duplicate types, i.e. if there is morethan one vertex of some type, we delete all but one. The number of siblings we keep isbounded by 2k – a function of the tree-depth.
We then proceed from the bottom up. At level i, we can again compute the type of avertex by looking at the children (whose number is bounded by a function of k) and theancestors in T , also bounded by k. We then again delete all duplicate siblings (and theirrespective subtrees). Again, the number of kept siblings is bounded by a function of k.
So overall the pruned tree-depth decomposition T ′ is a tree of height k with branchingbounded by a function of k. Thus its size is bounded by a function of the tree-depth ofG.
Note that we have considered every vertex of G once, with the number of steps at eachvertex bounded by a function of the parameter k. Thus the algorithm runs in linear FPTtime.
Remark. We say that the vertex that is kept when deleting vertices of the same typerepresents the deleted vertices. Note that the pruning defines a graph homomorphismϕ : V (T )→ V (T ′) that maps each vertex in T to the vertex of the same type in T ′ thatwas kept to represent it. It should be clear that the mapping can be computed whilepruning the tree.
Now we have all the tools to prove the following:
Theorem 4.5.10. Given a graph G, we can, in linear FPT time, find a b-box represen-tation of G, such that the boxicity of G is either b or b− 1.
Proof. Let k be the tree-depth of the input graph G. First we compute a tree-depthdecomposition T of G, and then use Lemma 4.5.9 to compute a pruned tree-depth de-composition T ′ of G with respect to T . Let G′ be the subgraph of G induced by thevertices of T ′. The size of G′ is bounded by a function of the tree-depth k of G, so we canfind a minimal b-box representation of G′ in time bounded by a function of k. Note thatb is at most the boxicity of G, since G′ is a subgraph of G.
Let (I ′1, . . . , I′b) be a minimal b-box representation of G′. We extend it to a (b+ 1)-box
representation of G. Recall that there is a homomorphism ϕ, computed while constructing
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T ′, that maps each vertex v ∈ G to the vertex ϕ(v) ∈ G′ that represents it in the prunedtree-depth decomposition. For each v ∈ G\V (G′) we add an interval for v that is identicalto the one for ϕ(v) to each I ′j. Let (I1, . . . , Ib) be the b-box representation obtained afteradding all the missing vertices. This can be computed in linear time.
This gives us all the edges from G, but possibly more than just those. We add anadditional interval graph J that removes unwanted edges. We define J recursively asfollows. Starting at the root of T , we define the interval of a subtree as an interval forthe root that contains the disjoint intervals of all the subtrees rooted at its children. Thisinterval graph is just the symmetric transitive closure of T . This can also be computedin linear time.
We now prove that (I1, . . . , Ib, J) is a (b + 1)-box representation of G. Suppose uv ∈E(G). Since T is a tree-depth decomposition of G, we have that uv ∈ E(J). Since ϕ is ahomomorphism, we have mapped u and v such that uv ∈ E(Ij) for all j. Thus the edgeis present in the box-representation.
Conversely, assume that uv is an edge in the box-representation. Since uv ∈ E(J), wemust have that u and v occur in the same branch, say without loss of generality that uis an ancestor of v in T . Moreover, we know that ϕ(u)ϕ(v) ∈ E(G′) and thus type(ϕ(v))records that it has an edge to a vertex in the same branch at the hight of ϕ(u). Since vand ϕ(v) agree on types, type(v) tells us that v is adjacent to u. Thus uv ∈ E(G).
This gives us an additive 1-approximation algorithm for Boxicity parameterized bytree-depth. So it is theoretically possible that the boxicity computed by the algorithmis off by one – but how likely is it to happen in practice? Unfortunately it is easy toconstruct simple examples where it does, see for example:
Example 4.5.11. Consider the two graphs in Figure 4.1: The right graph G′ is inducedby the vertices of a pruned tree-depth decomposition of the left graph G. The right graphG′ clearly is an interval graph, while the left G has no interval representation, i.e. boxicityof at least 2. It is easy to verify that the boxicity of G is indeed 2.
4.6 Open questions
What motivated us to study the problems in Section 4.5 is that tree-depth is a graphparameter that lies between vertex cover number and tree-width in the following sense:for any graph G we have tw(G) ≤ td(G) ≤ vc(G). The problems we studied in Section 4.5are known to be fixed-parameter tractable parameterized by vertex cover number, buteither known to be hard parameterized by tree-width or with an unknown status.
There are several problems with these characteristics that have an open status withregard to parameterizations with tree-depth. A group of problems with this property arethe following: Graph layout problems are optimisation problems where the objective isto linearly order the vertices to minimise some property. For example, the imbalance ofa vertex is the absolute value of the difference of the number of its neighbours to the leftand to the right. The Imbalance problem tries to find an order that minimises the sumof all the imbalances of vertices in a graph. Other graph layout problems are Cutwidth,Bandwidth and Distortion. Many graph layout problems have withstood attemptsto prove that they are fixed-parameter tractable parameterized by the tree-width of thegraph. However, it was shown by Fellows et al. [67] that a number of graph layoutproblems are fixed-parameter tractable parameterized by the vertex cover number of the
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Figure 4.1: An example where the boxicity of the pruned graph G′ is one less than the originalgraph G.
graph. There are also several graph colouring problems where parameterizations withtree-depth should be studied for the same reasons. (cf. [33, 69]).
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CHAPTER 5
Elimination distance
The study of parameterized algorithmics for graph problems has uncovered a large varietyof structural parameters of graphs. In Chapter 3 we have seen several examples of distanceparameters, specific kinds of structural graph parameters that measure the distance of agraph G to a graph class C in some way. Among these, the simplest distance parameteris ‘deletion distance to C’: it counts the number of vertices that one must delete from agraph G to obtain a graph in C (cf. Section 3.1).
In this chapter we introduce for any graph class C the graph parameter ‘eliminationdistance to C’. Elimination distance generalises the notion of deletion in the parameter‘deletion distance to C’. This is made precise below, but intuitively elimination distanceto a class C can be understood as a ‘parallelized deletion distance’ that allows deleting avertex from every connected component simultaneously at each deletion step until everycomponent is contained in a given class C. It is based on elimination trees or tree-depthdecompositions and can also be thought of as a natural generalisation of the parametertree-depth, which was introduced in Chapter 4. In fact, tree-depth can be characterised asthe elimination distance to the class of edgeless graphs, so it is a specific case of eliminationdistance. (See Section 5.4 below for a discussion of the relationship).
In Section 5.1 we formally introduce the notion of elimination distance to a class C inall generality and study the relationship to other parameters.
Recall from Section 3.2 that the vertex cover number of a graph G is the size of aminimal vertex cover of G. Alternatively, it is easy to see that it can be characterised asthe deletion distance to the class of edgeless graphs: if we delete every vertex from thegraph that is in the vertex cover, there are no edges in the graph left. So vertex covernumber can be seen as a special case of deletion distance to a class C.
A vertex cover can be also interpreted as a stricter version of a tree-depth decompo-sition, where the tree-depth decomposition has to be a path. Or, equivalently, just asin our description of elimination distance above, the notion of deletion in tree-depth canbe understood as a ‘parallelized deletion distance’ that generalises the notion of deletionin deletion distance: instead of deleting a fixed number of vertices, in a fixed number ofsteps we recursively delete vertices from each component of the graph until the graph hasno edges left.
In a sense, the graph parameter introduced in this chapter, elimination distance to aclass C, can be thought of as a natural generalisation of both deletion distance (generalisingthe notion of deletion) and tree-depth (allowing for more general classes than the classof edgeless graphs). The elimination distance to any class C of a graph G gives a lower
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Deletion distance to a class C Elimination distance to a class C
Vertex cover number Tree-depth
Figure 5.1: Illustration of the relationship between the parameters. An arrow means ‘is anupper bound for’.
bound for both the tree-depth of G and the deletion distance of G to C. It thus occupiesan interesting new space in the hierarchy of parameters demonstrated in Figure 5.1.
Tree-width is another parameter that gives a lower bound for both tree-depth andvertex cover number. In that sense, elimination distance to a class C is an alternativegeneralisation of vertex cover number and tree-depth. As a notion of distance it is morenatural than tree-width, because it is defined in terms of vertex deletions instead of the‘width’ of a tree decomposition. The relationship of elimination distance to C to tree-width depends on the choice of C: for example, if C is the class of graphs with degreebounded by d (for d ≥ 3), then C has unbounded tree-width and the same is a fortioritrue of graphs with elimination distance k to C. On the other hand, if C is just the classof edgeless graphs, then elimination distance to C is tree-depth. The class of all paths hasunbounded tree-depth, but constant tree-width.
Elimination distance offers a number of algorithmically interesting characterisations.It can be defined through recursive deletions, as described, but also order theoreticallyby defining an order on the vertices of the graph that tells us which vertices should be‘eliminated’ first. There is also a more combinatorial way: If a graph has eliminationdistance k to a class C, then it can be split in a graph with a tree-depth decompositionof height k, and a graph where every component is contained in C and connected to atmost one branch of the tree-depth decomposition. This allows for interesting algorithmicapplications exploiting the tree structure of the tree-depth decomposition. Just as fortree-depth, this can also be translated into an order theoretic definition. Lastly the classof graphs with elimination distance k to a class C can also be obtained by repeatedlyclosing C under the addition of apex vertices and taking the closure under disjoint unions.We discuss these alternative characterisation in Section 5.2.
In Section 5.3 we establish that if a class C has bounded expansion, then the classof graphs Ck with elimination distance k to C also has bounded expansion. This is doneusing one of the alternative characterisations established before.
Every choice of the graph class C gives us a different parameter. This makes theparameter very versatile, because the elimination distance to C has different propertiesfor different classes C. In Section 5.4 we consider the case where C is the class of edgelessgraphs and establish that this gives us an alternative characterisation of tree-depth.
In Section 5.5 we explore the case where C is a class of graphs of bounded degree.Along the way we establish a different characterisation of this particular parameter that
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is interesting in itself: we define the notion of an elimination order to degree d, whichis similar to the elimination order to C introduced in Section 5.2, but makes use of thelocality afforded by bounded degree classes. It also defines a partial order on the vertices,now indicating the order in which they need to be eliminated to reduce the degree to d,but is defined in a different way.
As discussed in the introduction, there are generally two reasons why the study of aparameter can be interesting in parameterized complexity. Either the parameter itself isof interest, or we want to understand the complexity of other problems with regard to theparameter. Elimination distance to a class C is an interesting parameter in both theseways and Chapter 6 and Chapter 7 give examples for each.
In Chapter 6 we show that Graph Isomorphism is FPT parameterized by elimina-tion distance to bounded degree. This generalises known results, in particular the resultby Bouland et al [23] that shows it is FPT parameterized by tree-depth and generalisedtree-depth. Here the parameter gives us a wider class of graphs where the Graph Iso-morphism problem is known to be tractable, and improves our understanding of theproblem.
In Chapter 7, we show that given a finite set of minors that characterise a minor-closed class C, we can find a finite set of minors that characterizes the class of graphs withelimination distance to C. This gives an explicit FPT-algorithm to check whether a givengraph has a certain elimination distance to C.
5.1 Elimination distance to a class CIn this section we formally introduce the parameter in its full generality. We relax thenotion of ‘deletion distance’ so that rather than considering the sequential deletion of kvertices, we consider the recursive deletion of vertices in a tree-like fashion. To be precise,we say that a graph G has elimination distance k+ 1 from a class C if, in each connectedcomponent of G we can delete a vertex so that the resulting graph has elimination distancek to C.
We want to make sure that deleting vertices gets us closer to the target class C, andthat is why we restrict the classes C to ones that are closed under taking subgraphs.Formally, elimination distance is defined as follows:
Definition 5.1.1. Let C be a class of graphs closed under taking subgraphs. The elimi-nation distance to C of a graph G is defined as follows:
edC(G) :=
0, if G ∈ C;
1 + minedC(G \ v) | v ∈ V (G), if G 6∈ C and Gis connected;
maxedC(H) | H a connected component of G, otherwise.
Remark. Observe that if a graph G has elimination distance 1 to a class D, and all graphsin D have elimination distance k to a class C, then G has elimination distance k+ 1 to C.This fact is going to be used in some induction proofs.
In the following we see the relationship of the parameter to the other distance param-eters deletion distance to a class C (Proposition 5.1.2) and tree-depth (Proposition 5.4.1).
For deletion distance, the following is a straightforward consequence that can be easilyseen from the definition. In the worst case we have to delete all vertices from a single
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connected component without being able to split the graph into several components. Inthat case deletion distance and elimination distance are the same.
Proposition 5.1.2. Let C be a class of graphs. If a graph G has deletion distance k toC, then G has elimination distance at most k to C.
Recall the notion of sparsity called ‘bounded expansion’ from Definition 2.3.31. InSection 5.3 we establish that if a class C has bounded expansion, then the class of graphsCk with elimination distance k to C also has bounded expansion. In order to prove this, wefirst we need to make use of a different characterisation of elimination distance introducedin in Section 5.2.
5.2 Alternative characterisations of elimination dis-
tance
In this section we introduce a number of algorithmically interesting characterisations ofelimination distance. First we give a characterisation given by an order on the vertices thattells us which vertices should be removed first. Next we establish a more combinatorialway that splits the graph in a part with a tree-depth decomposition of height boundedby the parameter, and one where every component is contained in C and connected to atmost one branch of the tree-depth decomposition. This allows for interesting algorithmicapplications exploiting the tree structure of the tree-depth decomposition. The actualdefinition is also given in order-theoretic terms. Lastly we establish a characterisationusing apex graphs and closures under disjoint unions.
5.2.1 Elimination order to CWe now introduce an equivalent characterisation of this parameter by defining an orderon the vertices. If G is a graph that has elimination distance k to a class C, then we canassociate a certain tree order ≤ with it as defined below. The idea is that from everyconnected component S of G we can delete a vertex vS so that the elimination distance toC is reduced. We define vS ≤ u for all vertices u ∈ S and vertices in distinct componentsof S \ vS are made incomparable with respect to ≤. We proceed in this fashion until allcomponents are contained in C, and make all vertices in these incomparable (and indeed≤-maximal). Note that all vertices that are not ≤-maximal must be comparable to theirneighbours. In the following we give the definition inspired by this and prove it gives usan equivalent notion to the one given in Definition 5.1.1:
Definition 5.2.1. Let G be a graph and let C be a class of graphs closed under takingsubgraphs. A tree order ≤ on V (G) is an elimination order to C for G if for each v ∈ V (G)we have that:
• if there is a vertex u > v, then there is no vertex w that is a neighbour of v andincomparable to v, i.e. there is no vertex w such that uw ∈ E(G) and v 6≤ w andw 6≤ v.
• if v is ≤-maximal, then the maximal subgraph S of G that only contains verticesthat are ≤-maximal and contains v is contained in the class C. Moreover any two
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vertices that are less than a vertex in S must be comparable, i.e. for any two verticesw,w′ ∈ S, we have that if u < w, u′ < w′ then u ≤ u′ or u′ ≤ u.
Remark. Note that this definition naturally extends the notion of an elimination orderdefined in Definition 2.4.9.
The following proposition implies Proposition 4.2.3, which states that the height ofan elimination order (see Definition 2.4.9) gives us an alternative characterisation of tree-depth.
Proposition 5.2.2. A graph G has edC(G) ≤ k if, and only if, there is an eliminationorder to C of height at most k for G.
Proof. Suppose first that edC(G) ≤ k. We proceed by induction on k. If k = 0, thenall components of G are contained in C and we define the elimination order ≤ to bethe identity relation on V (G). Then every v ∈ V (G) is maximal, and the connectedcomponent S of the subgraph of G induced by the ≤-maximal vertices of G containing vis thus contained in the class C.
Suppose k > 0 and the statement is true for smaller values. If G is not connected, weapply the following argument to each component. So in the following we assume that Gis connected. Thus, there is a vertex a ∈ V (G) such that the components C1, . . . , Cr ofG \ a all have edC(Ci) ≤ k− 1. So by the induction hypothesis each Ci has an eliminationorder ≤i to C of height at most k − 1 with the properties in Definition 5.2.1.
Let≤ := (a, w) | w ∈ V (G) ∪
⋃i
≤i .
Then ≤ is clearly a tree order for G. Note that there is no vertex w that is a neighbourof a and incomparable to a.
Conversely assume there is an elimination order ≤ to C of height d for G. We againproceed by induction on k. If k = 0, then ≤ is empty and all vertices v in G are ≤-maximal. Thus all components of G are contained in C as required.
Now, suppose k > 0. If v is an ≤-minimal element then for any u for which thereis a path from v to u, we must have v ≤ u. Thus, if G is connected, there is a unique≤-minimal element v. Note that ≤ restricted to a component C of G \ v has height k− 1and thus by the induction hypothesis we have that edC(C) ≤ k−1. If G is not connected,this argument can be applied to each component of G.
5.2.2 Separation into two parts
In the following we demonstrate that the previous characterisation can also be interpretedin a different way, giving us a second alternative characterisation of elimination order toC. We derive that characterisation by separating the maximal elements from the non-maximal elements: that way, we can split a graph with an elimination order to someclass C in two parts; one with an elimination order defined on it, and one where everyconnected component is contained in C. Furthermore, any two vertices from the formerpart that are neighbours of vertices from the same component of the latter part must becomparable in the elimination order. This means that if G is a graph that has eliminationdistance k to a class C, we can associate an elimination order ≤ for a subgraph H of G ofheight k with G, so that each component of G \V (H) is contained in C, and is connectedto H along just one branch (this is defined more formally below).
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Proposition 5.2.3. Let C be a class of graphs closed under taking subgraphs. Let G be agraph and let ≤ be an elimination order to C for G of height k. If A is the set of verticesin V (G) that are not ≤-maximal, then:
1. ≤ restricted to A is an elimination order of height k − 1 for G[A];
2. every component of G \ A is contained in C;
3. and if C is the vertex set of a component of G\A, and u, v ∈ A are ≤-incomparable,then either E(u,C) = ∅ or E(v, C) = ∅.
Proof. By Definition 5.2.1 ≤ is a tree-order. Moreover, since any v ∈ A is non-maximal,also by Definition 5.2.1, there is no vertex that is a neighbour of v and incomparable tov. Hence if there is an edge between any u, v ∈ A, either u < v or v < u, and (1) follows.
Since G \ A contains the ≤-maximal elements, it follows from the definition of anelimination order to C that every component of G \A is contained in C, establishing (2).
To show (3), let C be the vertex set of a component of G \A and let u, v ∈ A be suchthat E(u,C) 6= ∅ and E(v, C) 6= ∅. Then there are a, b ∈ C such that au, bv ∈ E(G).By Definition 5.2.1, since u is non-maximal and a is a neighbour of v, they must becomparable. Since a is maximal, we must have u < a. Similarly, v < b. So again byDefinition 5.2.1 we must have either u ≤ v or v ≤ u and we conclude that u and v arecomparable, proving (3).
We also have a converse to the above in the following sense.
Proposition 5.2.4. Let C be a class of graphs closed under taking subgraphs. Suppose Gis a graph with A ⊆ V (G) a set of vertices and ≤A an elimination order of G[A] of heightk, such that:
1. every component of G \ A is contained in C;
2. if C is the vertex set of a component of G \A, and u, v ∈ A are incomparable, theneither E(u,C) = ∅ or E(v, C) = ∅.
Then, ≤A can be extended to an elimination order to C for G of height k + 1.
Proof. Let
≤ :=≤A ∪ (v, v) | v ∈ (V (G) \ A)∪ (u, v) | u ∈ A, v ∈ C,C a component of G \ A,
E(w,C) 6= ∅ for some u ≤ w.
Then it is easily seen that ≤ is a tree order on G. Indeed, ≤A is, by assumption, a treeorder on A and for any v ∈ V (G) \ A, assumption (2) guarantees that w | w ≤ v islinearly ordered.
Let v ∈ V (G). If there is a vertex u > v, then by our construction v ∈ A. Since ≤Ais an elimination order, v is comparable to all its neighbours in A. Let w ∈ V (G) \ A bea neighbour of v not contained in A, and let C be the component of G \A containing w.Then by the construction we have u < w because w ∈ E(v, C) and thus E(v, C) 6= ∅.
Now suppose v is ≤-maximal, and let C be the connected component of G \ A con-taining v. By assumption (1), we have that C is contained in C. Let w,w′ ∈ C and let
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Figure 5.2: Illustration of an elimination order to C of a graph G. The tree on top is a tree-depth decomposition of a subgraph of G. The remaining components (at the bottom) are allcontained in C and connected to at most one branch of the tree-depth decomposition.
u, u′ be vertices of G such that u < w and u′ < w′. By the construction of ≤, theremust be vertices a ≥ u and a′ ≥ u′ such that E(a, C) 6= ∅ and E(a′, C) 6= ∅. So by thecontrapositive of assumption (2) we have that a and a′ are comparable. But ≤ is a treeorder, so we must have that u and u′ are comparable, completing the proof.
Remark (1). In the remainder of the dissertation, given a graph G and an eliminationorder ≤ to C for some class C we call the subgraph of V (G) induced by the non-maximalelements of the order ≤ the non-maximal subgraph of G under ≤.
Remark (2). In the proof of Proposition 5.2.4 above, a suitable tree order on a subsetA of V (G) is extended to an elimination order to C of G by making all vertices not inA maximal in the order. This is a form of construction we use repeatedly when we areworking with these definitions in Chapter 6.
Proposition 5.2.3 and Proposition 5.2.4 establish a third characterisation of eliminationdistance to a class C via a separation of the graph. Note that, although the definitionhere is given in terms of an elimination order, we could also state it in terms of a tree-depth decomposition: then the graph is split into a graph of tree-depth k, and one whereevery component is contained in C and connected to at most one branch of the tree-depthdecomposition. This is illustrated in Figure 5.2
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This separation can be exploited in algorithms, and we see an example of how it canbe used in Section 5.3, and later in Chapter 6.
5.2.2.1 Characterisation via games
Another immediate consequence of the characterisation via the separation in two parts,is that elimination distance to a class C can also be characterised via games. Recall fromSection 4.2 that tree-depth can be characterised via a Cop and Robber game, where theRobber player loses when she cannot move anymore. Similarly, the generalised tree-depthof a graph (Section 4.4.1) is defined via a Cop and Robber game, where the Robberplayer loses when she is confined to a path (with cops at the endpoints) or a cycle that isdisconnected (by cops) from the rest of the graph.
From the above it follows that the same Cop and Robber game can be adapted tocharacterise elimination distance to a class C. This is done by changing the winningcondition of the Cop player to confining the Robber player to a connected componentthat is part of C. The Cop player can then play the same strategy as for tree-depth onthe non-maximal subgraph of G under ≤.
5.2.3 Characterisation using apex graphs and closures underdisjoint unions
In this section we establish a third alternative characterisation of the class of graphs withelimination distance k to a class C, using apex graphs and closures under disjoint unions.
First we give the necessary definitions and establish notation. The closure of a binaryrelation R on a set X is the transitive relation R+ on set X such that R+ contains R andR+ is minimal
Definition 5.2.5. Let G,G′ be graphs. The disjoint union of G and G′, denoted GtG′,is the graph with vertex set
V (G)× 0 ∪ V (G′)× 1
and edge set
(u, 0)(v, 0) | uv ∈ E(G) ∪ (u, 1)(v, 1) | uv ∈ E(G′).
Definition 5.2.6. Let C be a class of graphs. The closure of C under taking disjointunions is the minimal class of graphs C that contains C, and has the additional propertythat for any two graphs G,G′ ∈ C there is H ∈ C with H ∼= G tG′.
The special case of a graph with deletion distance 1 to a class C is called an apex graphover C in this chapter.
Definition 5.2.7. We say that a graph G is an apex graph over a class C of graphs ifthere is a vertex v ∈ V (G) such that the graph G \ v ∈ C. The class of all apex graphsover C is denoted Capex. We say Capex is an apex graph class over C.
Remark. We will sometimes refer to such a vertex v as apex vertex, or just apex.
The characterisation of elimination distance introduced in this section is in terms ofthe iterated closure of C under the operation of disjoint unions and taking the class ofapex graphs. We introduce a piece of notation for this in the next definition.
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Definition 5.2.8. For a class of graphs C, let C0 := C, and Ci+1 := Ciapex.
We show next that the class Ck is exactly the class of graphs at elimination distancek from C.
Proposition 5.2.9. Let C be a class of graphs and k ≥ 0. Then Ck is the class of allgraphs with elimination distance at most k to C.
Proof. We prove this by induction. Only the graphs in C have elimination distance 0 toC, so the statement holds for k = 0.
Suppose the statement holds for k ≥ 0. If G ∈ Ck+1, then G is a disjoint union ofgraphs G1, . . . , Gs from Ckapex, so we can remove at most one vertex from each of the Gi
and obtain a graph in Ck. Thus the elimination distance of G to Ck is 1, and by inductionthe elimination distance to C is k + 1.
Conversely, if G has elimination distance k+1 to C, then we can remove a vertex fromeach component of G to obtain a graph G′ with elimination distance k to C. Using theinduction hypothesis each component of G′ is in Ck, and thus G ∈ Ck+1.
5.3 Elimination distance & classes of bounded expan-
sion
In this section we use one of the alternative characterisations established in the last sectionto prove that if C is a class of graphs with bounded expansion (cf. Definition 2.3.31), thenthe class of graphs with fixed elimination distance to C also has bounded expansion.
Proposition 5.3.1. Let C be a class of graphs with bounded expansion and let d ≥ 0.Then the class of graphs with elimination distance d to C also has bounded expansion.
Proof. Let C be a class of graphs with bounded expansion. We use the characterisationof bounded expansion classes via low tree-depth colourings (see Definition 2.3.32).
Recall from Theorem 2.3.33 that C has bounded expansion if, and only if, for everyinteger p there is an integer Np such that for any graph G ∈ C we have χp(G) ≤ Np. Weuse the characterisation of elimination distance defined above in Section 5.2.2 to split Ginto two parts and find a low tree-depth colouring for each of them.
Fix an integer p. Then there is an integer N such that for any graph G ∈ C we haveχp(G) ≤ N . Let G′ be a graph with elimination distance d to C. By Proposition 5.2.3there is an elimination order ≤ on G′ such that ≤ is an elimination order of height d on thegraph T induced by the non-≤-maximal vertices. All components of the graph U inducedby the ≤-maximal vertices are contained in C. Thus there is a tree-depth colouring ofU using at most N colours. Observe that it is not a problem to reuse colours from onecomponent in another, because the components are not connected.
We colour the non-≤-maximal vertices as follows: We use d fresh colours to colourevery level (see Definition 2.4.8) of ≤ in one of the d colours. Thus every colour inducesa subgraph of T with tree-depth just 1, and every combination of j ≤ d colours inducesa subgraph with tree-depth j.
Combining the colourings of both parts gives us a colouring of G′ that uses N + dcolours. Note that if S is a set of colours, and S ′ is a set of colours obtained by addingto S one of the d colours used for the non-≤-maximal vertices, then the tree-depth of the
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subgraph induced by S ′ is at most one more than the tree-depth of the subgraph inducedby S. Thus every set of colours S with |S| ≤ p induces a subgraph of G′ with tree-depthat most |S|, and therefore χp(G
′) ≤ N + d.So for every integer p, and every graph G′ with elimination distance d to C, there is
an integer Np (N + d above) such that χp(G′) ≤ Np. This shows that the class of graphs
Cd with elimination distance d to C has bounded expansion.
5.4 Elimination distance to the class of edgeless graphs
The class of edgeless graphs is the class of all the graphs where every vertex is isolated.Here we consider the elimination distance to that class and show that this gives us yetanother alternative characterisation of tree-depth, underlining the versatility of the pa-rameter.
If C is the class of edgeless graphs, then Definition 5.1.1 just yields the definition ofthe tree-depth of G (cf. Definition 4.0.4). So we have the following.
Proposition 5.4.1. Let C be the class of edgeless graphs and let k > 0. Let G be a graphwith at least one vertex. Then G has elimination distance k − 1 to C if, and only if, Ghas tree-depth k.
Proof. We prove this by induction on k. Let k = 1. A graph G has elimination distance0 to C if all its components are contained in C. Since C is the class of edgeless graphs thismeans none of the components contain an edge. So the tree-depth of G is 1. Conversely,if td(G) = 1, then all the vertices of G are isolated, and thus edC(G) = 0.
Let k > 1 and assume the statement holds for smaller values. If G has eliminationdistance k− 1 to C, there is a vertex v ∈ V (G) such that G \ v has elimination distancek − 2 to C. By the induction assumption G \ v has tree-depth k − 1 and thus Ghas tree-depth k. The converse is similar: If G has tree-depth k, then there is a vertexw ∈ V (G) such that G \ w has tree-depth k− 1. By the induction assumption G \ whas elimination distance k − 2 to C and thus, by definition, edC(G) = k − 1.
Remark. We just exclude the empty graph because in that case both tree-depth andelimination distance to the class of edgeless graphs are 0.
From that perspective elimination distance is a natural generalisation of tree-depth.In general elimination distance to a class C seems to be a much more powerful parameterthan tree-depth; for example we show in Chapter 8 that tree-depth is first-order definable.It is doubtful that the same holds for elimination distance to any class C, or even to anynon-trivial class – but it remains an open question.
From Proposition 5.4.1 it also follows that the tree-depth of a graph is an upper boundfor the elimination distance to any class C:
Proposition 5.4.2. Let C be a (non-empty) class of graphs closed under taking subgraphs.If a graph G has tree-depth k, then G has elimination distance at most k to C.
Proof. Let C be a class of graphs closed under taking subgraphs that contains at leastone graph. Let G be a graph with td(G) = k. If k = 0, G is the empty graph and sinceC is closed under taking subgraphs we have G ∈ C and thus G has elimination distance 0to C.
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If k > 0, by Proposition 5.4.1 we have that G has elimination distance k − 1 to theclass of edgeless graphs. Thus G has elimination distance at most k to the class containingonly the empty graph. Since C contains the class containing only the empty graph, thisproves that G has elimination distance to C at most k.
5.4.1 Elimination distance & generalised tree-depth
In Section 4.4 we discussed the parameter generalised tree-depth, introduced by Boulandet al. [23]. Generalised tree-depth generalises both tree-depth and max leaf number inthe sense that it is a lower bound for both. What is the relationship of this parameter toelimination distance?
Recall from Section 4.4.1 that the generalised tree-depth of a graph is defined via aCop and Robber game, where the Robber player loses when she is confined to a path(with cops at the endpoints) or a cycle that is disconnected (by cops) from the rest of thegraph. We will see that this gives us almost the elimination distance to paths and cycles(i.e. graphs of degree at most 2). However, since it is not enough to isolate the robber ona path, and the robber is only caught after a cop is placed at each end point, there is anadditional restriction to consider.
Consider for example a graph G that is a disjoint union of paths. Clearly G haselimination distance 0 to the class C2 of graphs of degree at most 2. However, the rulesof the game that defines generalised tree-depth force the Cop player to place two copson the path where the Robber player puts the robber. The generalised tree-depth of Gis thus 2. Note however that this is the worst case: It is never more than 2 and can beequal. So generalised tree-depth is ‘almost’ the elimination distance to C2.
In general if we have a graph G with elimination distance k to C2 there is an eliminationorder ≤ to C2 of height k on G by Proposition 5.2.3. The Cop player can use the order asa strategy so that after k rounds, there are only components of maximal degree at most2 left. If there are paths with no cops on them at this stage the Cop player might needtwo more rounds to win the game. Conversely, the strategy of the Cop player defines anelimination order to C2. It follows that the elimination distance to C2 is a lower bound forgeneralised tree-depth, and generalised tree-depth can be at most 2 more than eliminationdistance to C2:
Proposition 5.4.3. Let C2 be the class of graphs of degree at most 2. Then for any graphG we have the following:
edC2
(G) ≤ gtd(G) ≤ edC2
(G) + 2.
In the next section we explore the more general case where C is a class of boundeddegree. We choose a different way to define an elimination order and show that it givesus an alternative characterisation of elimination distance to classes of bounded degree.
5.5 Elimination distance to bounded degree classes
In this section we study elimination distance to graph classes of bounded degree. Theclass of graphs with degree bounded by a constant d is interesting, because many problemsbecome tractable when restricted to it. A very prominent example is graph isomorphism,which is an important application and the subject of Chapter 6. Figure 1.1 shows therelationship of elimination distance to bounded degree to other distance parameters.
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By defining C to be graphs of degree bounded by a constant d, we obtain a specialcase of Definition 5.1.1. To be less verbose, we define the elimination distance to degree dof a graph G, abbreviated edd(G), to mean the elimination distance to the class of graphswith maximal degree d.
In this section we introduce another characterisation of this parameter; if a graph haselimination distance k to degree d, we define an order equivalent to the elimination orderto C, where C is the class of graphs with maximal degree d. So this definition is similarto that given in Definition 5.2.1, but there is an important difference: because maximaldegree is a local property we do not need to refer to components here and instead state thedefinition in local terms of neighbourhoods of incomparable vertices. This is the definitionwe work with in Chapter 6.
Just as in Definition 5.2.1, the idea is that if G is a connected graph we can choosea vertex v so that deleting it strictly reduces the elimination distance to degree d. Wethen make v ≤ u for all vertices u and vertices in distinct components of G \ v are madeincomparable with respect to ≤. We proceed in this fashion until the remaining verticeshave degree at most d (and are incomparable, and indeed ≤-maximal). Note that anycomponent of the remaining vertices has maximal degree d, and the vertices of G \ Cthat are neighbours of vertices in C must be linearly ordered by ≤ and they all precedethe vertices of C in the order ≤. Thus, if we associate with every vertex v, the set Sv ofneighbours of v that are ≤-incomparable with v we can note the following: Sv has at mostd elements for any v; if Sv is not empty, then v must be ≤-maximal; and if u ∈ Sv thenu and v must have the same ≤-predecessors. This can be taken as the defining propertyof an elimination order to degree d, as below.
Definition 5.5.1. A tree order ≤ on V (G) is an elimination order to degree d for G iffor each v ∈ V (G) the set
Sv := u ∈ V (G) | uv ∈ E(G) and u 6≤ v and v 6≤ u
satisfies either:
• Sv = ∅; or
• v is ≤-maximal, |Sv| ≤ d, and for all u ∈ Sv, we have w | w < u = w | w < v.
Remark. Note that if Sv = ∅ for all v ∈ V (G), then an elimination order to degree d isjust an elimination order, in the sense of Definition 2.4.9.
Proposition 5.5.2. A graph G has edd(G) ≤ k if, and only if, there is an eliminationorder to degree d of height k for G.
Proof. Suppose first that edd(G) ≤ k. We proceed by induction on k. If k = 0, then Ghas no vertex of degree larger than d and we define the elimination order ≤ to be theidentity relation on V (G). Then every v ∈ V (G) is maximal, we have |Sv| ≤ d, and forall u ∈ Sv we have w | w < u = ∅ = w | w < v.
Suppose k > 0 and the statement is true for smaller values. If G is not connected, weapply the following argument to each component. So in the following we assume that Gis connected. Thus, there is a vertex a ∈ V (G) such that the components C1, . . . , Cr ofG \ a all have edd(Ci) ≤ k − 1. So by the induction hypothesis each Ci has a tree order
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≤i to degree d of height at most k − 1 with the properties in Definition 5.5.1. For eachv ∈ V (Ci) define
Siv := u ∈ V (Ci) | uv ∈ E(G) and u 6≤i v and v 6≤i u.
Let≤ := (a, w) | w ∈ V (G) ∪
⋃i
≤i .
Then ≤ is clearly a tree order for G. Note that Sa = ∅. Let v ∈ V (G) \ a be a vertexdifferent from a, say v ∈ V (Ci). Note Siv = Sv. If Sv 6= ∅, then v is ≤i-maximal, and thusalso ≤-maximal. Moreover, |Siv| = |Sv| ≤ d. Lastly for any u ∈ Sv:
w | w < u = a ∪ w | w <i u = a ∪ w | w <i v = w | w < v.
Conversely assume there is an elimination order ≤ to degree d of height k for G. Weagain proceed by induction on k. If k = 0, then ≤ is empty and all vertices v in G are ≤-maximal. Thus, Sv contains all neighbours of v and therefore v has at most d neighbours.In other words, ∆(G) ≤ d as required.
Now, suppose k > 0. If v is an ≤-minimal element then for any u for which thereis a path from v to u, we must have v ≤ u. Thus, if G is connected, there is a unique≤-minimal element v. Note that ≤ restricted to a component C of G \ v has height k− 1and thus by the induction hypothesis we have that edd(C) ≤ k−1. If G is not connected,this argument can be applied to each component of G.
Recall that Proposition 5.2.3 and Proposition 5.2.4 allow us to split a graph with anelimination order to degree d in two parts: one of low degree, and one with an eliminationorder defined on it.
The alternative characterisation in terms of the elimination order to degree d estab-lished above, and the ability to split the graph, are algorithmically very useful. As anapplication, in Chapter 6, we use these to establish that Graph Canonisation (and thusGraph Isomorphism) is FPT parameterized by elimination distance to bounded degree.This is done by constructing a canonical elimination order to degree d of G, based onan elimination order of a graph we call the torso of G, which contains the high-degreevertices of G, along with some additional edges.
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CHAPTER 6
Graph Isomorphism
The Graph Isomorphism (GI) problem is the problem of determining, given a pairof graphs G and H (as adjacency matrices), whether they are isomorphic, i.e., whetherthere is a bijective function ϕ : V (G) → V (H) such that uv ∈ E(G) if and only ifϕ(u)ϕ(v) ∈ E(H). The precise definitions can be found in Section 2.3.1.
More informally, GI asks whether the two given graphs are structurally equal, i.e. ifwe can rename the vertices of G to obtain H, or whether given a drawing of G we canrearrange the nodes in a way that gives us H. For example, the following is a mappingfrom the graph in Figure 6.1 to the graph in Figure 6.2 that preserves the edge relationin both directions:
a 7→ 1
b 7→ 6
c 7→ 8
d 7→ 3
g 7→ 5
h 7→ 2
i 7→ 4
j 7→ 7
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Figure 6.1: A graph.
Figure 6.2: Another graph.
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The problem is formally defined as follows:
Graph IsomorphismInput: Two graphs G and H.Problem: Are G and H isomorphic?
GI is clearly in NP (an isomorphism is a succinct certificate). For a problem in NP ithas an unusual status in complexity theory: it is neither known to be in P nor knownto be NP-complete – one of the few natural problems for which this is the case; see thediscussion below.
A naıve approach to solve GI would try out all of the n! possible mappings betweenV (G) and V (H) and would take exponential time. Despite considerable effort1, progresson the problem has been slow. However, while the problem has not been satisfactorilyresolved, the intense study of the problem produced a large number of algorithmic tech-niques and new concepts in complexity theory. Until recently the best known algorithmwas due to Babai and Luks [12] and runs in time 2
√n logn. However, it seems the situation
has changed: Very recently Babai [11] announced an algorithm that solves graph isomor-phism in quasi-polynomial time, i.e. in time O(2logc n) for some constant c > 1. Both theseresults are mostly based on group theoretic approaches.
This puts GI almost in the class P, but obstacles remain. The problem is very closelyrelated to the Group Isomorphism problem, which is in fact a special case. GroupIsomorphism asks whether two groups (G, ·) and (H, ), given in the form of theirCayley tables, are isomorphic, i.e. if there is a bijection ϕ : G → H such that for alla, b ∈ G we have ϕ(a) ϕ(b) = ϕ(a · b). In the 1970s Tarjan2 and Lipton et al. [116]independently observed that the Group Isomorphism problem can be solved in timenlogn+O(1) – exploiting the fact that a group of size n is generated by log n generators.However, despite significant efforts to improve this ‘trivial’ bound, and despite the muchricher structure available for groups, there has not been any real progress on the GroupIsomorphism problem. This seems to be the current frontier for the general GraphIsomorphism problem.
Despite the lack of success in showing that the problem is contained in P, it seemsunlikely that it is NP-hard. GI is known to be contained in co-AM and in level Lp2 of the lowhierarchy [13, 146], which means that if it were NP-hard, then the polynomial hierarchywould collapse to its second level, which most complexity theorists do not believe to belikely. This is the reason for the widespread belief that GI is in fact contained in P – amajor open problem at the moment.
GI is also one of the few ‘natural’ candidates3 for NP-intermediate problems. Theclass NP-intermediate contains the problems in NP that are neither in P nor NP-hard.Ladner [111] used a diagonalization based technique to show that such problems exist ifP 6= NP; in fact, he proves there are infinitely many. However, despite this fact there arenot many natural problems that are potential members of NP-intermediate. Most naturalproblems in NP can either be solved in polynomial time (and are thus in P) or can beshown to be NP-hard.
1The problem has been called “Isomorphism Disease” because of the time wasted by researchers[84, 139].
2Tarjan communicated the result in private correspondence.3The other problem that is usually mentioned in this context is Factoring. There are a few more
candidates for NP-intermediate problems depending on where one decides to draw the line between naturaland artificial.
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Graph Isomorphism has a large number of applications in practice. One of the prob-lems that initiated the practical study of the problem was the need to identify canonicalnames for molecules to find related work on chemical compounds [62]. Other applicationsinclude the analysis of circuits in hardware design [136], optical character recognition[156], the semantic web [34], improving the PageRank algorithm [20, 105], autonomousrepair of sensor networks [37], graph mining of social networks [150], de-anonymizingsocial networks [14], and even fighting terrorism [90] and paleontology [144]. The mostsignificant application is possibly the elimination of symmetries in instances of the booleansatisfiability problem (SAT) [8], that is used to speed up industrial SAT solvers. High-performance SAT solvers have an increasingly important role in solving combinatorialproblems in industry.
Despite its uncertain status in complexity theory, GI can be considered solved for mostof these practical applications. Powerful graph isomorphism software, such as nauty byMcKay [126, 127], Traces by Piperno [127, 137], saucy by Darga et al. [46] and conautoby Lopez-Presa [119] can quickly solve most instances that show up in applications. Allthese programs are based on the same idea: they compute a canonical labeling of thegraph, employing several heuristics to speed up the process.
Despite the fact that researchers have not found a polynomial-time algorithm for GIon general graphs, polynomial-time algorithms are known for a variety of special classes ofgraphs. Most of the published research on the problem over the last 40 years is concernedwith this question. Many of these lead to natural parameterizations of GI by means ofstructural parameters of the graphs which can be used to study the problem from thepoint of view of parameterized complexity. For instance, it is known that GI is in XP (cf.Definition 2.2.7) parameterized by the genus of the graph [70, 128], by maximum degree[12, 120] and by the size of the smallest excluded minor [138], or more generally, thesmallest excluded topological minor [93].
For each of the parameters mentioned in the last paragraph it remains an open questionwhether Graph Isomorphism is FPT. On the other hand, GI has been shown to beFPT when parameterized by eigenvalue multiplicity [60], tree distance width [158], themaximum size of a simplical component [151, 153] and minimum feedback vertex set [108].Bouland et al. [23] showed that the problem is FPT when parameterized by the tree-depthof a graph and extended this result to a parameter they termed generalised tree depth(cf. Section 5.4.1) that also proves that the problem is FPT parameterized by max leafnumber. In a recent advance on this, Lokshtanov et al. [118] have shown that graphisomorphism is also FPT parameterized by tree-width.
The result we present in this chapter extends the results of Bouland et al. and isincomparable with that of Lokshtanov et al. We show that graph canonisation is FPTparameterized by elimination distance to degree d, for any constant d. The structuralgraph parameter we use is an instance of elimination distance introduced in Chapter 5,which is a general paradigm that is also of interest in the context of other graph problems.It should be noted that the class of all graphs with degree bounded by d (for d ≥ 3) hasunbounded tree-width and the same is a fortiori true of graphs with elimination distancek to degree d. It should also be noted that generalised tree depth in is an upper boundfor the special case of elimination distance to degree 2 (cf. Section 5.4.1).
To motivate the choice of elimination distance, and the choice of the class of graphsof bounded degree, in the case of GI, consider the simplest notion of distance for a graphG to a class where GI is tractable: the number k of vertices of G that must be deleted
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to obtain a graph with no edges (cf. Section 3.1). This is, of course, just the size of aminimal vertex cover in G (cf. Section 3.2) and is a parameter that has been extensivelystudied (see for instance [67]). Indeed, it is also quite straightforward to see that GI isFPT when parameterized by vertex cover number.
Consider two ways this observation might be strengthened. The first is to relax thenotion of what we consider to be tractable, i.e. to consider a greater class of graphs whereGI is still tractable. For instance, as there is, for each d, a polynomial time algorithmdeciding GI among graphs with maximum degree d, we may take this as our base case,i.e. as the class of graphs we measure the distance from. We then parameterize G by thenumber k of vertices that must be deleted to obtain a subgraph of G with maximum degreed. This yields the parameter deletion distance to bounded degree, which we consider inSection 6.2 below.
Alternatively, we can relax the notion of “distance” to that of “elimination distance”introduced in Chapter 5, so that rather than considering the sequential deletion of kvertices, we consider the recursive deletion of vertices in a tree-like fashion.
In our main result of this chapter, established in Section 6.3, we combine these twoapproaches by parameterizing G by the elimination distance to the class of graphs withmaximal degree d (or just elimination distance to degree d). We show that, for any fixedd, this gives a structural parameter on graphs for which graph canonisation is FPT.
We begin by recalling some result for isomorphism on bounded degree graphs in thenext section.
6.1 Isomorphism on bounded-degree graphs
In this section we collect some well known results about isomorphism tests and canoni-sation of bounded degree graphs that we use below. Luks [120] shows that isomorphismof bounded-degree graphs is decidable in polynomial time. It is well-known that thisresult extends, by an easy reduction, to coloured graphs of bounded-degree. Though thisreduction is folklore, for the sake of completeness, we present it here explicitly.
Proposition 6.1.1. The isomorphism problem for coloured graphs of maximum degreed can be reduced to the isomorphism problem of graphs of maximum degree d + 2 inpolynomial time.
Proof. Let G,G′ be graphs and let c, c′ : V (G) → 1, . . . , k be colourings of G,G′
respectively for some k ∈ N.We define H to be the graph whose vertices include V (G) and, additionally, for each
v ∈ V (G), c(v) + 1 new vertices uv1, . . . , uvc(v)+1. The edges of H are the edges E(G) plus
additional edges so that the vertices v and uv1, . . . , uvc(v)+1 form a simple cycle of length
c(v) + 2. We obtain H ′ in a similar way from G′.We claim that G ∼= G′ if, and only if, H ∼= H ′. Clearly, if G ∼= G′ and ϕ is an
isomorphism witnessing this, it can be extended to an isomorphism from H to H ′ bymapping uvi to u
ϕ(v)i . For the converse, suppose H ∼= H ′ and let ϕ : H → H ′ be an
isomorphism. We use it to define an isomorphism ϕ′ from G to G′. Note that, if v ∈ V (G)is not an isolated vertex of G, then it has degree at least 3 in H. Since ϕ(v) has the samedegree, it is in V (G′), and we let ϕ′(v) = ϕ(v). If v is an isolated vertex of G, thenits component in H is a simple cycle of length c(v) + 2. The image of this componentunder ϕ is a simple cycle of H ′ which must contain exactly one vertex v′ of V (G′). We let
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ϕ′(v) = v′. It is easy to see that there is an edge between v1, v2 in G if, and only if, thereis an edge between ϕ′(v1) and ϕ′(v2) in G′. To see that ϕ′ also preserves colours, note
that ϕ must map the cycle containing uvc(v) to the cycle containing uϕ′(v)c(ϕ′(v)) and therefore
c(v) = c(ϕ′(v)).Note that if G and G′ are graphs of maximum degree d, then H,H ′ are graphs of
maximum degree d+ 2.
As Luks [120] proves that isomorphism of bounded degree graphs can be decided inpolynomial time, we have the following:
Theorem 6.1.2. We can test in polynomial time whether two (coloured) graphs withmaximum degree bounded by a constant are isomorphic.
In 1983 Babai and Luks [12] give a polynomial-time canonisation algorithm for boundeddegree graphs. Just as above we can reduce canonisation of coloured bounded degreegraphs to the bounded degree graph canonisation problem.
Theorem 6.1.3. Let C be a class of (coloured) bounded degree graphs closed under iso-morphism. Then there is a canonical form F for C that allows us to compute F (G) inpolynomial time.
6.2 Deletion distance to bounded degree
We first study the notion of deletion distance to bounded degree and establish in thissection that graph isomorphism is FPT with this parameter. Though the result in thissection is subsumed by the more general one in Section 6.3.2 and also by a result ofKratsch and Schweitzer [108], it provides a useful warm-up. It also provides a specificalgorithm for the problem, while Kratsch and Schweizer [108, Theorem 1] solve the moregeneral problem of graph isomorphism given two graphs that are k vertex deletions awayfrom a graph class characterised by excluded induced subgraphs.
The notion of deletion distance to bounded degree is a particular instance of the generalnotion of distance to triviality introduced by Guo et al. [95]. So from that perspective,and in the context of graph isomorphism, we have chosen triviality to mean graphs ofmaximum degree at most d.
In the present warm-up we only give an algorithm for the graph isomorphism problem,though the result easily holds for canonisation as well (and this follows from the moregeneral result in Section 6.3.2).
Recall the notion of deletion distance to a class C introduced in Definition 3.1.1. Weintroduce some notation that allows us to be less verbose in this section.
Definition 6.2.1. We say that a graph G has deletion distance k to degree d if it hasdeletion distance k to the class of graphs of maximum degree d. We call a set of verticesv1, . . . , vk such that G \ v1, . . . , vk has maximum degree d a d-deletion set.
We make the following observations:
Remark. To say that G has deletion distance 0 from degree d is just to say that G hasmaximum degree d. Also note that if d = 0, then a d-deletion set is just a vertex coverand the minimum deletion distance the vertex cover number of G.
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We show that isomorphism is fixed-parameter tractable on graphs parameterized by kwith fixed degree d; in particular we give a procedure that computes in linear time a setU of vertices of size polynomial in k so that any deletion set must be found in U if oneexists. This then allows for exhaustive search over all potential isomorphisms. We callU a bounding set for the deletion set. The construction is reminiscent of a kernelizationresult in [64] for the problem of determining whether a graph has deletion distance k todegree d, though that does not yield a bounding set containing all deletion sets.
Theorem 6.2.2. For any graph G and integers d, k > 0, we can identify in linear time asubgraph G′ of G, a set of vertices U ⊆ V (G′) with |U | = O(k(k+ d)2) and a k′ ≤ k suchthat: G has deletion distance k to degree d if, and only if, G′ has deletion distance k′ to dand, moreover, if G′ has deletion distance at most k′, then any minimum size d-deletionset for G′ is contained in U .
Proof. Let A := v ∈ V (G) | deg(v) > k+ d. Now, if R is a minimum size d-deletion setfor G and G has deletion distance at most k to degree d, then |R| ≤ k and the verticesin V (G \ R) have degree at most k + d in G. So A ⊆ R. This means that if |A| > k,then G must have deletion distance greater than k to degree d and in that case we letG′ := G, k′ := k and U = ∅.
Otherwise let G′ := G \A and k′ := k− |A|. We have shown that every d-deletion setof size at most k must contain A. Thus G has deletion distance k to degree d if, and onlyif, G′ has deletion distance k′ to degree d.
Let S := v ∈ V (G′) | degG′(v) > d and U := S ∪ NG′(S). Let R′ ⊆ V (G′) be aminimum size d-deletion set for G′. We show that R′ ⊆ U . Let v 6∈ U . Then by thedefinition of U we know that degG′(v) ≤ d and all of the neighbours of v have degreeat most d in G′. So if v ∈ R′, then G \ (R′ \ v) also has maximal degree d, whichcontradicts the assumption that R′ is of minimum size. Thus v 6∈ R′.
Note that the vertices in G′ \ (R′∪N(R′)) have the same degree in G′ as in G and thusall have degree at most d. So S ⊆ R′ ∪N(R′) and thus |U | ≤ k′+ k′(k+ d) + k′(k+ d)2 =O(k(k + d)2).
Finally, the sets A and U defined as above can be found in linear time, and G′, k′ canbe computed from A in linear time.
Remark. It may be noted that the set U is canonical in the sense that if there is anisomorphism from G to a graph H, this must take U to the corresponding set in H.But, this is not essential to our argument below. What is important is that U containsall possible deletion sets for G. In particular, if U = ∅ and k′ > 0, then there are nod-deletion sets of size at most k′.
Next we see how the bounding set U can be used to determine whether two graphs withdeletion distance k to degree d are isomorphic by reducing the problem to isomorphismof coloured graphs of degree at most d.
In the following suppose we are given two graphs G and H, with d-deletion setsS := v1, . . . , vk and T := w1, . . . , wk respectively. Further suppose that the mapvi 7→ wi is an isomorphism on the induced subgraphs G[S] and H[T ]. We can then testif this map can be extended to an isomorphism from G to H using Theorem 6.1.3. To beprecise, we define the coloured graphs G′ and H ′ which are obtained from G\S and H \Trespectively, by colouring vertices. A vertex u ∈ V (G′) gets the colour i | vi ∈ NG(u),i.e. the set of indices of its neighbours in S. Vertices in H ′ are similarly coloured by the
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sets of indices of their neighbours in T . It is clear that G′ and H ′ are isomorphic if, andonly if, there is an isomorphism between G and H, extending the fixed map between Sand T . The coloured graphs G′ and H ′ have degree bounded by d, so Theorem 6.1.3 givesus a polynomial-time isomorphism test on these graphs.
Now, given a pair of graphs G and H which have deletion distance k to degree d, let Aand B be the sets of vertices of degree greater than k + d in the two graphs respectively.Also, let U and V be the two bounding sets in the graphs obtained from Theorem 6.2.2.Thus, any d-deletion set in G contains A and is contained in A ∪ U and similarly, anyd-deletion set for H contains B and is contained in B ∪V . Therefore to test G and H forisomorphism, it suffices to consider all k-element subsets S of A∪U containing A and allk-element subsets T of B ∪ V containing B, and if they are d-deletion sets for G and H,check for all k! maps between them whether the map can be extended to an isomorphism
from G to H. As d is constant this takes time O∗((
k3
k
)2· k!)
, which is O∗(27k log k
). (See
the remark after Definition 2.2.3 for an explanation of the O∗ notation.)
6.3 Elimination distance to bounded degree
We now consider elimination distance to bounded degree and show in this section thatGraph Isomorphism is FPT with respect to this parameter. The result in this sectionsubsumes the result in Section 6.2.
Just as deletion distance in the previous section, the notion of elimination distance tobounded degree can also be interpreted as a particular instance of the notion of distanceto triviality introduced by Guo et al. [95]. From that perspective, and in the context ofgraph isomorphism, we have again chosen triviality to mean graphs of bounded maximumdegree. However, now we consider the more general elimination distance as the distance.
The main idea of the proof is to identify canonical structures in the graph. It iseasy to see that in a graph G that has elimination distance k to degree d, all vertices ofsufficiently high degree have to be ‘eliminated’, i.e. they have to be non-maximal elementsof the elimination order. However, it may be the case that in an optimal elimination ordersome low-degree vertices may also need to be non-maximal elements of the eliminationorder. A key idea in our proof is to show that G contains a canonically defined setof vertices C (which we call its torso), including the high-degree vertices so that if Ghas elimination distance k to degree d, then we can reduce the maximum degree to dby ‘eliminating’ just vertices in C, and moreover this yields an elimination order whoseheight is bounded by a function of k and d. This is established in Section 6.3.1.
With this characterisation established, we are able to present the canonisation al-gorithm in Section 6.3.2. The central technique used here is inspired by Lindell’s treecanonisation algorithm [115]. We define an order v on isomorphism types of colouredgraphs equipped with an elimination order recursively on the height of this order, withthe base case being a canonical order on coloured graphs of bounded degree. We thenshow that given a graph G, separated into its torso C and a low-degree part, we canconstruct an elimination order for G (i.e. a tree-depth decomposition of C) which is v-minimal. We use the result of Bouland et al. [23] on canonisation of bounded tree-depthgraphs to bound the branching in the search and establish that canonisation is FPT.
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6.3.1 Canonical separation into high and low degree parts
The aim of this section is to show that if a graph G has elimination distance k to degree d,then there is an elimination order to degree d whose height is still bounded by a functionof d and k and in which the set of non-maximal elements is canonical. To be precise, weidentify a graph which we call the d-degree torso of G, which contains all the vertices of Gof degree more than d and has additional edges to represent paths between these verticesthat go through the rest of G. We show that this torso necessarily has tree-depth boundedby a function of k and d and an elimination order witnessing this can be extended to anelimination order to degree d of G.
The result is established through a series of lemmas. A pattern of construction thatis repeatedly used here is that we define a certain set A of vertices of G and constructan elimination order of G[A]. It is then shown that extending the order by making allvertices in V (G) \ A maximal yields an elimination order to degree d of G. Necessarily,in this extended order, all the non-maximal elements are in A.
We first do this for graphs of degree bounded by k + d. To be precise, the followinglemma establishes that if G has elimination distance k to degree d and moreover themaximum degree of G is at most k + d, then we can construct an alternative eliminationorder to degree d on G in which all the vertices of degree greater than d are included inthe non-maximal subgraph and the height of the new elimination order to degree d is stillbounded by a function of k and d.
Lemma 6.3.1. Let G be a graph with maximum degree ∆(G) ≤ k + d. Let ≤ be anelimination order to degree d of height k of G with non-maximal subgraph H, and letA = V (H) ∪ v ∈ V (G) | degG(v) > d. Then G has an elimination order v to degree dof height at most k(k + d+ 1) for which A is the set of non-maximal elements.
Proof. Let G,H,A and ≤ be as in the statement of the lemma. We adapt ≤ to anelimination order v of G[A].
For each w ∈ A \ V (H) let Cw be the component of G \ V (H) that contains w. Notethat N(Cw) 6= ∅, because deg(w) > d, so at least one vertex in H must be adjacent tow. By Definition 5.5.1, the vertices in N(Cw) are linearly ordered by ≤. We write b(w)to denote the unique ≤-maximal element of N(Cw) for each w ∈ A \ V (H). For eachb ∈ V (H), let Wb := w ∈ A \ V (H) | b(w) = b, and let vb be an arbitrary linear orderon Wb.
For any u, v ∈ V (G), define u v v if one of the following holds:
• u = v;
• u, v ∈ H and u ≤ v;
• u ∈ H, v ∈ G \ A and u ≤ v;
• u ∈ H, v ∈ A \ V (H) and u ≤ b(v);
• u ∈ A \ V (H), v ∈ G \ A and b(u) ≤ v;
• u, v ∈ A \ V (H), b(u) = b(v) and u vb v.
It follows from the construction that v restricted to A is an elimination order of G[A],and that v is an elimination order to degree d of G.
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For each b ∈ V (H), the set v ∈ H | v ≤ b has at most k elements, by the assumptionon the height of the order ≤. Since G has maximum degree k + d and Wb ⊆ N(v ∈ H |v ≤ b), we have that Wb has at most k(k + d) vertices. Since the height of any v-chainis at most the height of a ≤-chain plus |Wb|, we conclude that the height of v is at mostk(k + d+ 1).
The lemma above allows us to re-arrange the elimination order so that it includesall vertices of large degree. In contrast, the next lemma gives us a means to re-arrangethe elimination order so that all vertices of small degree are made maximal in the order.This is again achieved while keeping the height of the elimination order bounded by afunction of k and d. Note that while Lemma 6.3.1 only applies to graphs of maximaldegree bounded by k + d, the following result applies to general graphs.
Lemma 6.3.2. Let G be a graph. Let A be the set containing all vertices of degree greaterthan d, i.e. A = v ∈ V (G) | degG(v) > d, and let ≤ be an elimination order to degree dof G of height k with non-maximal subgraph H, such that H contains A. Then, there isan elimination order to degree d of G of height at most k((k + 1)d)2
k+ 1 for which A is
the set of non-maximal elements.
Proof. Let G,H,A and ≤ be as in the statement of the lemma. We assume that G isconnected – if not, we can apply the argument to each component of G. We construct anelimination order v of G[A] from ≤, making sure that it has height at most k((k+1)d)2
k,
and satisfying the conditions of Proposition 5.2.4. This then extends to an eliminationorder to degree d of G by making all vertices not in A maximal, as in that Proposition.
Let J := H \ A. For v ∈ V (J), let Kv be the set of vertices w ∈ A such that:
1. v ≤ w;
2. there is a path from v to w through G \ A; and
3. for any u with u < v, there is no path from u to w through G \ A.
Note that because ≤ is a tree order and due to the third condition, the sets Kv are pairwisedisjoint. Let K := A \ (
⋃v∈V (J)Kv) be the set of vertices in A that are not contained in
Kv for any v.For each v ∈ V (J), let vv be an arbitrary linear order on Kv. The idea behind the
construction below is that we replace v in the elimination order by Kv, ordered by vv.Formally, for any u,w ∈ V (G), define u v w if one of the following holds:
• u = w;
• u ∈ Kv, w ∈ G \ A and v ≤ w;
• u ∈ K, w ∈ G \ A and u ≤ w;
• u,w ∈ Kv and u vv w;
• u ∈ Kv, w ∈ Kv′ and v < v′;
• u ∈ K, w ∈ Kv and u ≤ v;
• u ∈ Kv, w ∈ K and v ≤ w;
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• u,w ∈ K and u ≤ w.
We first show that v is an elimination order for G[A]. The construction ensures v isa tree order. Let u,w ∈ A. We show that if u ≤ w, then either u v w or w v u. Wego through all possible cases: If u = w, we have u v w. If there is some v ∈ V (J) suchthat u,w ∈ Kv, then u v w or w v u. Suppose u ∈ Kv, w ∈ Kv′ for v, v′ ∈ V (J). Bydefinition of Kv, v ≤ u and similarly v′ ≤ w. Since we also have u ≤ w, by the tree orderproperty either v ≤ v′ or v′ ≤ v and thus u v w or w v u. If u ∈ K and w ∈ Kv, thenboth u, v ≤ w, so either u ≤ v or v ≤ u, and thus either u v w or w v u. The case whereu ∈ Kv, w ∈ K is symmetric. Finally, if both u,w ∈ K, then u v w. Thus if uw ∈ E(G),we have u ≤ w or w ≤ u and therefore u v w or w v u. Hence v is an elimination orderfor G[A].
Let Z be a component of G \ A. By assumption A is the set of all vertices of degreegreater than d in G, and therefore Z has maximum degree d.
Suppose u, v ∈ A are two vertices that are connected to Z, i.e. EG(u, V (Z)) 6= ∅ 6=EG(v, V (Z)). We show that either u v v or v v u. Note that there is a path P throughZ ⊆ G \A from u to v, i.e. all vertices in P , except for the endpoints, lie outside of A. IfP contains no vertices from J , then the connected component Z ′ of G \ V (H) containingP \ u, v satisfies EG(u, V (Z ′)) 6= ∅ 6= EG(v, V (Z ′)) and thus u ≤ v or v ≤ u, andtherefore by the above u v v or v v u. Otherwise, P contains vertices from J . Let w bea ≤-minimal vertex in V (P ) ∩ V (J). Then there is a path outside of A from w to u, andalso to v (both part of P ). Moreover, if neither u ≤ v nor v ≤ u, then w ≤ u and w ≤ v.Thus u and v are in Kw (or in Kw′ for some w′ < w), and therefore u v v or v v u.
It remains to show that the size of Kv is bounded by k((k + 1)d)2k
for all v ∈ V (J).Let G′ be the graph obtained from G by adding an edge between two vertices s, t ∈ V (J)whenever there is a path through G \V (H) between s and t. This increases the degree ofvertices in V (J) by at most kd, because each of these vertices is connected to at most dcomponents of G \ V (H) and each of these is connected to at most k vertices in H. Nowthere is a path between two vertices s, t ∈ J in G outside of A if, and only if, there is apath between s and t in G′[V (J)]. Moreover, ≤ is also an elimination order for G′[V (J)].So, as G′[V (J)] has tree-depth at most k, by Lemma 4.1.1 it does not contain a path oflength more than 2k. Since each vertex on the path has degree at most (k + 1)d, we canreach at most ((k + 1)d)2
kvertices in A on paths only containing vertices outside of A.
Thus |Kv| ≤ ((k + 1)d)2k
and the height of v is bounded by k|Kv| ≤ k((k + 1)d)2k.
Next we introduce the notion of d-degree torso and prove that it captures the propertiesthat we require of an elimination order to degree d, that is, we show that an eliminationorder of the torso can be extended to an elimination order to degree d of the whole graph.
Definition 6.3.3. Let G be a graph, let d > 0 and let H be the subgraph of G induced bythe vertices of degree larger than d. The d-degree torso of G is the graph obtained fromH by adding an edge between two vertices u, v ∈ H if there is a path through G \ V (H)from u to v in G, i.e. a path where all vertices on the path, except for the endpoints, lieoutside of H.
Lemma 6.3.4. Let G be a graph and let C be the d-degree torso of G. Let H = G[V (C)]and let ≤ be an elimination order for H. Then ≤ is an elimination order for C of heighth if, and only if, ≤ can be extended to an elimination order to degree d for G of heighth+ 1.
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Proof. Let G,C,H and ≤ be as above.Suppose ≤ is an elimination order for C. Since C is a supergraph of H, this means
that ≤ is an elimination order for H. Let Z be a component of G \ V (H). Since Ccontains all vertices of degree greater than d, Z has maximum degree d. If E(Z, u) 6= ∅and E(Z, v) 6= ∅ for two vertices u, v ∈ H, then there is a path through Z ⊆ G \ V (H)connecting u and v, so by the definition of the d-degree torso uv ∈ E(C) and thus u, vare ≤-comparable. We can extend ≤ to a tree order ≤′ on V (G) where all the verticesfrom V (G) \ V (H) are maximal.
Conversely assume that ≤ can be extended to an elimination order to degree d forG. Let uv ∈ E(C). If uv ∈ E(H), then u and v must be ≤-comparable. Otherwiseuv 6∈ E(H), so there is a path through G \ V (H) from u to v in G, i.e. both u and vare connected to a component Z of G \ V (H) and thus comparable. Therefore ≤ is anelimination order for C.
The next lemma establishes an upper bound on the tree-depth of the torso of a graphwhen the maximum degree is bounded.
Lemma 6.3.5. Let G be a graph with elimination distance k to degree d and maximumdegree ∆(G) ≤ k + d. Let C be the d-degree torso of G and let ≤ be a minimum height
elimination order for C. Then ≤ has height at most k(k+d+1)((k(k+d+1)+1)d)2k(k+d+1)
.
Proof. Let v be a minimum height elimination order to degree d of G. Since G haselimination distance to degree d at most k, the height of v is at most k. Let H be thenon-maximal subgraph of G under v and define
A = V (H) ∪ v ∈ V (G) | degG(v) > d.
By Lemma 6.3.1, the graph G[A] has an elimination order of height at most k(k+d+1)that can be extended to an elimination order to degree d for G.
Let A′ = v ∈ A | degG(v) > d. By Lemma 6.3.2, the graph A′ has an elimination
order ≤ of height at most k(k + d+ 1)((k(k + d+ 1) + 1)d)2k(k+d+1)
that can be extendedto an elimination order to degree d for G.
Lastly note that A′ = V (C), so that by Lemma 6.3.4, ≤ is an elimination order forC.
We are now ready to prove the main result of this section, which establishes that anelimination order of the d-degree torso can be extended to an elimination order to degreed of the whole graph, with height bounded by a function of only k and d.
Theorem 6.3.6. Let G be a graph that has elimination distance k to degree d. Let ≤ bea minimum height elimination order of the d-degree torso G. Then ≤ can be extended toan elimination order to degree d of G of height at most
k((k + 1)(k + d))2k
+ k(1 + k + d)(k(1 + k + 2d))2k(1+k+d)
+ 1.
Proof. We show that the d-degree torso of G has an elimination order of height at mostk((k + 1)(k + d))2
k+ k(1 + k + d)(k(1 + k + 2d))2
k(1+k+d). The theorem then follows by
Lemma 6.3.4.Let C be the (k + d)-degree torso of G. We first show that the tree-depth of C is
bounded by k((k + 1)(k + d))2k. To see this, let v be an elimination order to degree d of
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G of minimum height with non-maximal subgraph H. Note that H contains all vertices ofdegree greater than k+d, because vertices in G\V (H) are adjacent to at most k verticesin H.
Let A = v ∈ V (H) | degG(v) > k + d. By Lemma 6.3.2, the graph G[A] has anelimination order of depth at most h := k((k + 1)(k + d))2
kthat can be extended to
an elimination order to degree k + d of G of height h + 1. Note that A = V (C), so byLemma 6.3.4, the order is an elimination order for C. Let ′ denote its extension toG.
Let Z be a component of G\A and let CZ be the d-degree torso of Z. By Lemma 6.3.5,there is an elimination order Z for CZ of height at most k(k + d + 1)((k(k + d + 1) +
1)d)2k(k+d+1)
. Let vZ be the -maximal element in C such that there is a w ∈ CZ withvZ ′ w. Define
≤′:= ∪⋃Z
Z ∪⋃Z
(v, w) | v ′ vZ , w ∈ CZ.
Observe that C ∪⋃Z CZ has as a subgraph the d-degree torso of G. Thus ≤′ is an
elimination order for the d-degree torso of G. The height of ≤′ is bounded by
td(C) + maxtd(CZ)Z
which in turn is bounded by
k((k + 1)(k + d))2k
+ k(1 + k + d)(k(1 + k + 2d))2k(1+k+d)
.
6.3.2 Graph canonisation parameterized by elimination distanceto bounded degree
In this section we show that Graph Canonisation, and thus Graph Isomorphism,is FPT parameterized by elimination distance to bounded degree. The algorithm is basedon Lindell’s algorithm [115] that decides isomorphism of trees by establishing an orderingon them. We extend a version of this algorithm used by Bouland et al. [23] to decideisomorphism on graphs of bounded tree-depth.
Recall from Definition 2.3.16 that a canonical form F for a class of graphs C is anendomorphism of C that assigns a graph G to a graph isomorphic to G and that assignstwo isomorphic graphs G ∼= H to the same graph, i.e. G ∼= F (G) = F (H). The GraphCanonisation for a class C is the problem of computing the canonical form of a givengraph from C. The Graph Canonisation problem for a class C is at least as hard asGraph Isomorphism on that class, because it is easy to see that we can use an algo-rithm that solves the Graph Canonisation problem to solve Graph Isomorphism.However, it is not known whether the reverse also holds.
The central idea is to define a total pre-order, i.e. a total, reflexive and transitivebinary relation, v on graphs (G,≤) equipped with an elimination order to degree d,with the property that (G,≤) v (G′,≤′) and (G′,≤) v (G,≤′) if, and only if, (G,≤)is isomorphic to (G′,≤′). We then exhibit an algorithm that constructs, given a graphG, the elimination order to degree d on G that makes (G,≤) v-minimal. This servesas a canonical representative of G. Before defining v formally, we need some additionalterminology.
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Definition 6.3.7. Let G be a graph. We denote the number of connected componentsof G by #G.
The definition of v is by induction on the height of the order ≤. Inductively, we needto consider coloured graphs and so formally, vk is a pre-order on triples (G,χ,≤) whereχ : V (G)→ X is a colouring of G and ≤ is an elimination order to degree d of G.
Definition 6.3.8. For a graph G, with a colouring χ : V (G)→ X and a vertex v ∈ V (G),we write χv for the colouring of G \ v given by χv : V (G \ v) → X × 0, 1 whereχv(u) = (χ(u), 1) if (u, v) ∈ E(G) and χv(u) = (χ(u), 0) if (u, v) 6∈ E(G).
By Theorem 6.1.3 we can find a canonical form of each coloured graph Z of degreeat most d in polynomial time. This allows us to define a pre-order (see Definition 2.4.1)v0 on the graphs by taking the lexicographical order of the string-representation of thecanonical forms. That order is clearly reflexive and transitive, but since isomorphic graphsshare the same canonical form, it is not antisymmetric – so it is a pre-order and not apartial order. In other words, that pre-order v0 is such that for every pair of Z,Z ′ of suchgraphs, either Z v0 Z
′ or Z ′ v0 Z and Z ∼= Z ′ if, and only if, both Z v0 Z′ and Z ′ v0 Z.
In the following definition we extend this to a pre-order v on coloured graphs with anelimination order to degree d by induction on the height of the order.
Definition 6.3.9. Let G,G′ be graphs with colourings χ and χ′ and respective eliminationorders ≤,≤′ to degree d. We say that (G,χ,≤) v (G′, χ′,≤′) if one of the following holds:
1. height(≤) < height(≤′);
2. height(≤) = height(≤′) = 0 and (G,χ) v0 (G′, χ′);
3. height(≤) = height(≤′) > 0 and #G < #G′;
4. height(≤) = height(≤′) > 0, #G = #G′ = 1 and χ(v) < χ′(v′) where v, v′ areminimal elements of G,G′ with respect to ≤,≤′ respectively;
5. height(≤) = height(≤′) > 0, #G = #G′ = 1 and
(G \ v, χv,≤) v (G′ \ v′, χ′v′ ,≤′),
where v, v′ are minimal elements of G,G′ with respect to ≤,≤′ respectively;
6. height(≤) = height(≤′) > 0, s = #G = #G′ > 1, and there is an enumerationG1, . . . , Gs of the components of G and G′1, . . . , G
′s of the components of G′ such
that
(a) whenever i ≤ j, (Gi, χ,≤) v (Gj, χ,≤) and (G′i, χ′,≤′) v (G′j, χ
′,≤′); and
(b) if for any i, (Gi, χ,≤) 6v (G′i, χ′,≤′), then there is a j < i such that (G′j, χ
′,≤′) 6v(Gj, χ,≤).
To see that this inductive definition is well-founded, note that the recursive use of v ison graphs either equipped with an elimination order to degree d of strictly smaller height(clause (5)) or with strictly fewer components (clause (6)). The next two propositions areaimed at showing that this definition establishes a linear pre-order on coloured graphswith an elimination order that classifies them up to isomorphism.
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Proposition 6.3.10. For any pair (G,χ,≤), (G′, χ′,≤′) of coloured graphs with an elim-ination order to degree d, at least one of (G,χ,≤) v (G′, χ′,≤′) or (G′, χ′,≤′) v (G,χ,≤)holds.
Proof. The proof is by induction on the height k of ≤ and for a fixed k, by induction on thenumber of components. If k = 0, then either height(≤′) > 0 and (G,χ,≤) v (G′, χ′,≤′)by clause (1) of the definition, or height(≤′) = 0 and at least one of (G,χ,≤) v (G′, χ′,≤′)or (G′, χ′,≤′) v (G,χ,≤) must hold by clause (2) and the definition of v0.
Now suppose k > 0. If one of the first four conditions of Definition 6.3.9 is satisfied,we are done. So suppose they fail and suppose #G = #G′ = 1. Then, if v and v′ are theminimal elements of G,G′ with respect to ≤,≤′, the height of these orders on G \ v andG′\v′ is strictly less than k. By induction hypothesis, either (G\v, χv,≤) v (G′\v′, χ′v′ ,≤′)or (G′ \ v′, χv′ ,≤′) v (G \ v, χv,≤), so the claim follows.
Finally, suppose height(≤) = height(≤′) > 0 and s = #G = #G′ > 1 and we arguethat in all such cases, clause (6) ensures the result. By the induction hypothesis and theprevious case, for any pair of components Gi and Gj of G, one of (Gi, χ,≤) v (Gj, χ,≤)or (Gj, χ,≤) v (Gi, χ,≤) must hold and similarly for the components of G′. Thus, thecomponents of each can be ordered so that whenever i ≤ j, (Gi, χ,≤) v (Gj, χ,≤) and(G′i, χ
′,≤′) v (G′j, χ′,≤′). Also, by the induction hypothesis, we have for each i, either
(Gi, χ,≤) v (G′i, χ′,≤′) or (G′i, χ
′,≤′) v (Gi, χ,≤). If the first case holds for all i, then(G,χ,≤) v (G′, χ′,≤′). If the second case holds for all i, then (G′, χ′,≤′) v (G,χ,≤).If neither holds for all i, consider the least i such that either (Gi, χ,≤) 6v (G′i, χ
′,≤′) or(G′i, χ
′,≤′) 6v (Gi, χ,≤). In the former case, we have (G′, χ′,≤′) v (G,χ,≤) and in thelatter (Gi, χ,≤) v (G′i, χ
′,≤′). In all cases, the proposition is established.
This proposition implies that it is always possible to enumerate the componentsG1, . . . , Gs of G in v-order. The next proposition shows that this order is, essentially,unique.
Proposition 6.3.11. Let G,G′ be graphs with colourings χ and χ′ and let ≤,≤′ be elim-ination orders to degree d on G,G′ respectively.
Then both (G,χ,≤) v (G′, χ′,≤′) and (G′, χ′,≤′) v (G,χ,≤) if, and only if, (G,χ,≤) ∼=(G′, χ,≤′).
Proof. Suppose (G,χ,≤) ∼= (G′, χ,≤′) and suppose, without loss of generality that (G,χ,≤) v(G′, χ′,≤′). We aim to show that also (G′, χ′,≤′) v (G,χ,≤), and we do this by inductionon the height k of ≤ and for a fixed k, by cases on the number of components. If k = 0then since (G,χ) ∼= (G′, χ′), the conclusion follows from the definition of v0 and clause (2)of Definition 6.3.9. If k > 0 and #G = 1 then the induction hypothesis and the fact that(G \ v, χv,≤) ∼= (G′ \ v′, χ′v′ ,≤′) together imply that (G \ v, χv,≤) v (G′ \ v′, χ′v′ ,≤′) and(G′ \ v′, χ′v′ ,≤′) v (G \ v, χv,≤), so the proposition holds by clause (4) of Definition 6.3.9.Finally, suppose that k > 0 and #G > 1 and let G1, . . . , Gs enumerate the components ofG in v order. By the isomorphism between (G,χ,≤) and (G′, χ′,≤′) and the inductionhypothesis, it follows that for the corresponding enumeration G′1, . . . , G
′s of components
of G′ in v order, we have that (Gi, χ,≤) ∼= (G′i, χ,≤′) for each i. Then by clause (6) ofDefinition 6.3.9, both (G,χ,≤) v (G′, χ′,≤′) and (G′, χ′,≤′) v (G,χ,≤) hold.
For the other direction, suppose (G,χ,≤) v (G′, χ′,≤′) and (G′, χ′,≤′) v (G,χ,≤)If height(≤) < height(≤′), then (G′, χ′,≤′) 6v (G,χ,≤), so assume that height(≤) =height(≤′) = k and we proceed by induction on k. If k = 0, we have by clause (2)
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that (G,χ) v0 (G′, χ′) and (G′, χ′) v0 (G,χ) which, by definition of v0 means that(G,χ) ∼= (G′, χ), and the two orders are both trivial.
Suppose then that k > 0 and assume first that G and G′ are both connected. Thenboth G,G′ have minimal elements v, v′ with respect to ≤,≤′. By the induction hypothesiswe have (G\v, χv≤) ∼= (G′\v′, χv′ ,≤′), so let σ : V (G\v)→ V (G′\v′) be an isomorphismwitnessing this. Extend this to σ : V (G) → V (G′) by setting σ(v) = v′ and we claimthat this is an isomorphism between (G,χ,≤) and (G′, χ′,≤′). It is clear that σ takes theorder ≤ to ≤′ since v and v′ are minimal for these respective orders. Also, σ preservescolours since the colourings χv and χv′ refine the colourings χ and χ′ respectively and byclause (4) we have χ(v) = χ′(v′). Finally, we need to verify that σ preserves the edgerelation. Since σ preserves all edges, we only need to verify that uv ∈ E(G) if, and onlyif, σ(u)v′ ∈ E(G′). But, this follows from the fact that χv(u) = χ′v′(σ(u)).
Now, if G and G′ have different number of connected components, then we cannothave both (G,χ,≤) v (G′, χ′,≤′) and (G′, χ′,≤′) v (G,χ,≤). So, we are left with thecase where #G = #G′ = s > 1. Let G1 v · · · v Gs be the connected components of Gand let G′1 v · · · v G′s be the connected components of G′. Then by definition of v wehave that both (Gi, χ,≤) v (G′i, χ
′,≤′) and (G′i, χ′,≤′) v (Gi, χ,≤) for each i ≤ s. By
the argument for connected graphs above, we have (Gi, χ,≤) ∼= (G′i, χ′,≤′) for each i and
we are done.
Algorithm 1 Recursive construction of minimal elimination order to degree d.
Input: A graph G, a subgraph C ⊆ G, and `, d ∈ N.Output: A v-minimal elimination order to degree d of G, where all the sub-maximal
vertices are from C.1: procedure FindMinimalOrder(G, C, `, d)2: if #G > 1 then3: return
⋃i FindMinimalOrder(Gi, C[Gi ∩ C], `, d), where the Gi are the con-
nected components of G.4: else if ∆(G) ≤ d then5: return ∅.6: else7: find the set of potential roots R of a minimal elimination order on C.8: for w ∈ R do9: colour(u) := colour(u) ∪ ` for all neighbours u of w
10: <w:= FindMinimalOrder(G \ w,C \ w, `+ 1, d)11: <w:=<w ∪(w, u) | u ∈ V (G) \ w.12: return <w that makes (G,<w) v-minimal.
In order to define an algorithm for constructing a minimal elimination order to degreed, we need one further ingredient. For a graph G of tree-depth k, we say that a vertexv is a potential root of a minimal elimination order on G if td(G \ v) < k. We knowfrom Bouland et al. [23] that there is a computable function f such that the number ofpotential roots in any graph of tree-depth k is at most f(k). Moreover, the set of all suchvertices can be found by an algorithm running in time f(k)n3 where n is the number ofvertices in G.
Recall from Definition 2.4.8 that the level of an element v in a tree order ≤ is thelength of the chain w ∈ V (G) | w ≤ v.
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This now allows us to define Algorithm 1 which computes a minimal elimination orderto degree d. The algorithm is started with the graph G, its d-degree torso C, the level `initially set to 0 and the degree d. It then recursively chooses a potential root and returnsthe order that is minimal with respect to v.
With this, we are able to establish our main result.
Theorem 6.3.12. Graph Canonisation is FPT parameterized by elimination distanceto degree d, for any d.
Proof. Suppose we are given a graph G with |V (G)| = n. We first compute the d-degreetorso C of G in O(n4) time. Using Algorithm 1 we compute a v-minimal elimination order< to degree d on G. By the analysis from Bouland et al. [23, Theorem 11], this takestime O(h(k)n3log(n)) for some computable function h. Note for the base case that we cancompute the canonical representation of a component Z with ∆(Z) ≤ d by Theorem 6.1.3in polynomial time (where the degree of the polynomial depends on d).
Note that the order v also induces a natural order on the vertices of G and we canthus enumerate the vertices of G in a canonical way in linear time.
Since we can use the Graph Canonisation problem to solve Graph Isomorphism,this implies the following:
Corollary 6.3.13. Graph Isomorphism is FPT parameterized by elimination distanceto degree d, for any d.
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CHAPTER 7
Graph classes characterised byexcluded minors
This chapter establishes that the problem of determining elimination distance to anyminor-closed class C is FPT, and provides an explicit algorithm for the problem if we aregiven a set of forbidden minors characterising C.
A minor of a graph is obtained by deleting vertices and edges, and by contractingedges. A graph class is minor-closed if whenever some graph is a minor of a graph in theclass, that graph must also be contained in the class. (The definitions can be found inSection 2.3.2.)
For example the class of planar graphs, i.e. graphs that can be embedded into theplane without edges crossing, is clearly minor-closed. Wagner [155] showed that the classof planar graphs is exactly the class characterized by not having the complete graph K5
and the complete bipartite graph K3,3 as minors. A more general statement, for a longtime known as Wagner’s Conjecture, was that every minor-closed class is characterizedby such a finite set of ‘forbidden minors’.
In a series of twenty papers spanning over 500 pages from 1983 to 2004 Robertson andSeymour developed their theory of graph minors and concluded with a proof of Wagner’sconjecture, which is now called the Robertson-Seymour Theorem (cf. Theorem 2.3.21).The theorem states that the class of all undirected graphs, partially ordered by the graphminor relationship, is in fact well-quasi-ordered. This is equivalent to the statement thatevery minor-closed class of graphs is characterised by a finite set of ‘forbidden minors’.
Further examples of minor-closed classes include the class of all trees, the class of allforests, and for every value of k, the class of graphs with genus bounded by k (i.e. ageneralisation of planar graphs), the class of graphs with tree-depth bounded by k, theclass of graphs with path-width (or tree-width) bounded by k. For some of these theexplicit characterisation is known.
The Robertson-Seymour Theorem also has a very interesting consequence for the studyof algorithmics. Robertson and Seymour [142] show that there is a polynomial-timealgorithm1 that checks, given a graph G, whether a fixed graph H is a minor of G.This implies that for any fixed minor-closed class C of graphs, there is a polynomial time
1The runtime Robertson and Seymour originally showed was cubic, but it was recently improved toquadratic by Kawarabayashi et al. [104]. Also note that when checking whether H G, the runtime ispolynomial in the size of the graph G and has exponential dependence on the size of H.
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algorithm to test membership: given a graph G, just iterate over all the ‘forbidden minors’and check if one of them is a minor of G. If one of them is, G 6∈ C, otherwise G ∈ C.However, there is a huge constant cost in the runtime of the algorithm which makes itnot useful in practice.
In the area of parameterized complexity the technique based on the Robertson-Seymouralgorithm is often used to show that a problem is FPT, usually motivating further researchinto the problem revealing more efficient algorithms. The technique can be applied for agraph parameter k if the graphs with parameter ≤ k are minor-closed. For example, sinceboth the classes of tree-depth ≤ k and tree-width ≤ k are minor-closed, testing whethertree-depth or tree-width is at most k is FPT.
The catch is that this technique is non-constructive: if we do not know what the‘forbidden minors’ are, we know that there is a polynomial-time algorithm, but we cannotuse it. Although many graph classes are known to be minor-closed, the set of ‘forbiddenminors’ characterising them is usually not known. Since knowing the ‘forbidden minors’is a prerequisite for applying the algorithm from the Robertson-Seymour Theorem, thismakes the search for ‘forbidden minors’ an important research question.
In this chapter we show that for certain graph classes the ‘forbidden minors’ can becomputed. Recall the parameter elimination distance to a class C that we introduced inChapter 5. It is a distance parameter based on elimination trees or tree-depth decompo-sitions that is a lower bound for both tree-depth (cf. Section 5.4) and deletion distance(cf. Section 3.1).
Here we show that if a class of graphs C is minor-closed and its set of ‘forbiddenminors’ is known, then for every k > 0, we can also compute the set of ‘forbidden minors’that characterise the class of graphs with elimination distance k to C. By showing thatwe can compute the ‘excluded minors’ in this particular case, we establish that there isan explicit algorithm that can check whether a graph has elimination distance k to C ifthe ‘excluded minors’ for C are available to the algorithm.
Our result also implies that checking whether a given graph G has elimination distancek to a minor-closed class C is fixed-parameter tractable, answering a question posed in[28].
A large number of graph classes can be characterised by excluded minors, and an evenlarger number can be characterised by the elimination distance to such graph classes.Thus our result here gives a general method for a large number of problems and is thusan algorithmic meta-theorem.
In Chapter 8 we establish another algorithmic meta-theorem on the tractability ofgraph problems that are both slicewise first-order definable and slicewise nowhere dense.Note that our results discussed in Chapter 8 do not apply here: While a proper minor-closed class is always nowhere dense, it is not generally first-order definable (for instance,neither the class of acyclic graphs nor the class of planar graphs is), and eliminationdistance to such a class is also not known to be first-order definable. Thus, our resultsfrom Chapter 8 do not apply here.
7.1 Elimination distance to excluded minors
In this section we show that for any k > 0, the class of graphs with elimination distancek to a minor-closed class C is also minor-closed, and, moreover, if we have access tothe list of ‘forbidden minors’ characterising C, we can find the list of ‘forbidden minors’
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characterising the class of graphs with elimination distance k to C. We first pin down theexact computational problem that we are trying to solve and give an outline of the proofstrategy.
Recall the notion of elimination distance (Definition 5.1.1) from Chapter 5. For con-venience we restate the definition here.
Definition 7.1.1 (Definition 5.1.1). Let C be a class of graphs closed under taking sub-graphs. The elimination distance to C of a graph G is defined as follows:
edC(G) :=
0, if G ∈ C;
1 + minedC(G \ v) | v ∈ V (G), if G 6∈ C and Gis connected;
maxedC(H) | H a connected component of G, otherwise.
We build on work of Adler et al. [6] to show that from a finite list of the ‘forbiddenminors’ characterising C, we can compute the set of ‘forbidden minors’ characterising thegraphs with elimination distance k to C. Adler et al. show how to do this for apex graphs,from which one immediately obtains the result for graphs that are k deletions away fromC. To extend this to elimination distance k, we show how we can construct the forbiddenminors for the closure of a minor-closed class under disjoint unions.
In the following we show that determining the elimination distance of a graph toa minor-closed class C is FPT when parameterized by the elimination distance. Moregenerally, we formulate the following parameterized problem where the forbidden minorsof C are also part of the parameter.
Recall from Definition 2.3.20 that we write M(C) for the set of minimal excludedminors of C, i.e. the set of graphs in the complement of C such that for each G ∈ M(C)all proper minors of G are in C.
Elimination Distance to Excluded MinorsInput: A graph G, a natural number k ∈ N and a finite set of graphs MParameter: k +
∑H∈M |H|
Problem: Does G have elimination distance k to the class C characterised byM(C) = M?
In Section 7.2 we apply the characterisation of elimination distance k to a class Cestablished in Section 5.2.3. We first use this characterisation to establish that for anyminor-closed class C, the class of graphs which have elimination distance k to C is also aminor-closed class. We then show how to compute a set of excluded minors for the classof graphs which have elimination distance k to C from a set of excluded minors for C.
7.2 Computing the excluded minors of Capex and CIn this section we first show that for any minor-closed class C the class of graphs whichhave elimination distance k to C is also a minor-closed class. Then we give a procedure tocompute a set of excluded minors for the class of graphs which have elimination distancek to C from a set of excluded minors for C.
Recall the alternative characterisation of elimination distance k to a class C in termsof apex graphs and closures under disjoint unions that we established in Section 5.2.3.
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First we show that if C is a minor-closed class of graphs then so is the class of graphsCk with elimination distance k to C for any k ≥ 0.
Proposition 7.2.1. Let C be a minor-closed class. Then Ck is also minor-closed.
Proof. Indeed, it is well-known that Capex is minor-closed for any minor-closed C: con-tracting any edge, or removing any edge or vertex, leads to another graph in Capex. For,if G ∈ Capex with apex v ∈ V (G), then any graph obtained from G \ v by an edgecontraction or removal of a vertex is contained in C, and thus any graph obtained fromG by a contraction of an edge not incident to v, or a removal of a vertex other than v,gives us a graph in Capex. It is easy to see that the same holds for any edge removal orcontraction of an edge incident to v. And if v itself is removed, any other vertex may bechosen as the apex.
So it is just left to show that C is also minor-closed. But it is clear that if H is a minorof a graph G that is the disjoint union of graphs G1, . . . , Gs, then H itself is the disjointunion of (possibly empty) minors of G1, . . . , Gs. Thus, the class of graphs of eliminationdistance at most k to a minor-closed class C is itself minor-closed.
We next show how to construct a set of excluded minors of Ck from the correspondingset for C.
To obtain M(Ck), we need to iteratively compute M(Capex) and M(C) from M(C).Adler et al. [6] show that from the set of minimal excluded minors M(C) of a class C, wecan compute M(Capex) (and thus, in fact, the minimal set of excluded minors for the classof graphs with deletion distance k to C for any fixed value k).
Theorem 7.2.2 ([6], Theorem 5.1). There is a computable function that takes the set ofgraphs M(C) characterising a minor-closed class C to the set M(Capex).
We next aim to show that from M(C) we can also compute M(C). Together withTheorem 7.2.2 this implies that from M(C) we can compute M(Ck), the set of minimalexcluded minors for the class of graphs with elimination distance k to C.
We begin by characterising minor-closed classes that are closed under disjoint unionsin terms of the connectedness of their excluded minors.
Lemma 7.2.3. Let C be a class of graphs closed under taking minors. Then C is closedunder taking disjoint unions if, and only if, each graph in M(C) is connected.
Proof. Let C be a minor-closed class of graphs, and let M(C) = H1, . . . , Hs be its setof minimal excluded minors.
Suppose each of the graphs in M(C) is connected. Let H ∈ M(C) and let G =G1⊕· · ·⊕Gr be the disjoint union of graphs G1, . . . , Gr ∈ C. Because H is connected, wehave that H G if, and only if, H Gi for one 1 ≤ i ≤ r. So, since all the Gi ∈ C, wehave H 6 G and thus G ∈ C. This shows that C is closed under taking disjoint unions.
Conversely assume one of the graphs H ∈ M(C) is not connected and let A1, . . . , Atbe its components. Then A1, . . . , At ∈ C, since each Ai is a proper minor of H, and H isminor-minimal in the complement of C. However, A1 ⊕ · · · ⊕ At = H 6∈ C.
An important tool is the connection closure of a graph G, which defines an edge-minimal connected supergraph of G.
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Definition 7.2.4. For a graph G with connected components G1, . . . , Gr, let H denotethe set of connected graphs H with V (H) = V (G) such that the subgraph of H inducedby V (Gi) is exactly Gi. We define the connection closure of G to be the set of all minimal(under the subgraph relation) graphs in H. The connection closure of a set S of graphsis the union of the connection closures of the graphs in S.
Next we observe each of the graphs in the connection closure of a graph has the samenumber of edges.
Lemma 7.2.5. Let G be a graph with e edges and m connected components. Then anygraph in the connection closure of G has exactly e+m− 1 edges.
Proof. Let H be a graph in the connection closure of G. Then H has G as a subgraph, soe of its edges come from G. In addition, H is connected, so it has at least m−1 additionaledges corresponding to a tree on m vertices connecting the m components. Because His minimal (under the subgraph relation), it does not have more than m − 1 additionaledges. Thus H has exactly e+m− 1 edges.
Lemma 7.2.6. Let C be a minor-closed class of graphs. Then M(C) is the set of minor-minimal graphs in the connection closure of M(C).
Proof. Let C be a minor-closed class of graphs, with M(C) its set of minimal excludedminors, and let M be the connection closure of M(C).
Let G be a graph such that H 6 G for all H ∈ M . Suppose for a contradiction thatG is not a disjoint union of graphs from C. Then there is a component G′ of G that isnot in C and therefore there is a graph H ∈ M(C) such that H G′. We show that oneof the graphs in the connection closure of H is a minor of G′.
Let w1, . . . , ws be the vertex set of H and consider the image T1, . . . , Ts of the minormap from H to G′. Let T be a minimal subtree of G′ that contains all of the Ti. Sucha tree must exist since G′ is connected. Let H be the graph with the same vertex set asH, and an edge between two vertices wi, wj whenever either wiwj ∈ E(H) or when thereis a path between Twi and Twj in T that is disjoint from any Twk with wi 6= wk 6= wj.
We claim that H is in the connection closure of H. By construction, H is connected andcontains all components of H as disjoint subgraphs, so we only need to argue minimality.H has no vertices besides those in H so no graph obtained by deleting a vertex wouldcontain all components of H as subgraphs. To see that no edge of H is superfluous, weknow from Lemma 7.2.5 it has exactly e+m−1 edges and thus no proper subgraph couldbe connected and have all components of H as disjoint subgraphs. By the constructionH G′ G, so by the transitivity of the minor relation we have that H G.
Conversely let G be an arbitrary graph and assume that H ∈ M and H G. BecauseH is connected, there is a connected component G′ of G such that H G′. Now theremust be a graph H ∈ M(C) such that H is in the connection closure of H, and since His a subgraph of H, H H. Then, by the transitivity of the minor relation, H G′ andthus G′ 6∈ C. Therefore G is not a disjoint union of graphs from C.
7.3 Computing the excluded minors of CkNow we can put everything together and prove our main theorem:
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Theorem 7.3.1. There is a computable function which takes a set M of excluded minorscharacterising a minor-closed class C and k ≥ 0 to the set M(Ck).
Proof. The proof is by induction. For k = 0, the set of minimal excluded minors of C0 isM(C0) = M(C), which is given. For k > 0, we have that Ck = Ck−1apex. By the inductionhypothesis we can compute M(Ck−1), by Theorem 7.2.2 we can compute M(Ck−1apex)and using Lemma 7.2.6 we can compute the connection closure of M(Ck−1apex) to obtainM(Ck−1apex) = M(Ck).
So by the Robertson-Seymour Theorem (cf. Theorem 2.3.21) we have the following:
Corollary 7.3.2. The problem Elimination Distance to Excluded Minors isfixed-parameter tractable.
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CHAPTER 8
Slicewise first-order and slicewisenowhere-dense classes
In this chapter we establish a general technique, usually called an algorithmic meta-theorem, that allows us to show that for various classes C of sparse graphs, and severalmeasures of distance to such classes, the problem of determining the distance of a givengraph G to C is fixed-parameter tractable. More precisely, we establish that any problemthat is both slicewise nowhere dense and slicewise first-order definable (these terms aredefined precisely below in Definition 8.2.1 and Definition 8.2.2) is FPT. This includesproblems such as deletion distances, and more general edit distances to several sparsegraph classes.
In particular, this technique implies that if we have a class C that is first-order defin-able and nowhere dense and the distance measure we are interested in is also first-orderdefinable (that is to say, for each k there is a formula that defines the graphs at distance kfrom C), then the problem of determining the distance is FPT. More generally, if we havea parameterized problem (Q, κ) that is slicewise nowhere dense and slicewise first-orderdefinable, and a measure of distance to it is definable in the sense that for any valuesof k and d, there is a first-order formula defining the graphs at distance d to the classG | G ∈ Q and κ(G) ≤ k, then the problem of deciding whether a graph has distanceat most d to this class is FPT parameterized by k + d.
As a consequence of this it follows that if C is a nowhere dense class of graphs thatis definable by a first-order formula, then the parameterized problem of determining thedistance of a graph G to C is FPT, for various notions of distance that can be themselvesso defined.
As an application, we give new proofs of a number of existing results in the literature,which we explore in Section 8.3. We show that certain natural distance measures to sparsegraph classes are FPT. In particular, we get that various forms of edit distance to classesof bounded-degree graphs are FPT (a result established by Moser and Thilikos [129] fordeletion distance, by Mathieson [125] for several editing distances, and by Golovach [88]by more direct methods).
A simple example is deletion distance to graph classes of bounded degree. Consider,for each d > 0, the class of graphs with maximum degree at most d. These classes areboth first-order definable and nowhere dense. So our result implies that the problem ofdetermining whether a graph has degree at most d is FPT parameterized by d – however,
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it is easy to see that this problem is in fact in P. Here we are concerned with distanceparameters and we show in Section 8.3 that several natural distance parameters, suchas deletion distance are first-order definable. Moreover, we will see that the problem ofdetermining whether a graph has a certain deletion distance to a graph class with boundedmaximal degree is slicewise nowhere dense. This means that the problem ‘Can we removek vertices to obtain a graph with maximum degree less than d?’ is fixed-parametertractable.
Another interesting application is obtained by considering elimination distance of agraph G to the class C of edgeless graphs. Recall from Section 5.4 that this is nothingother than the tree-depth of G. While it is an open question whether elimination distanceto a class C is (or is not) in general first-order definable, it is in the particular case whereC is the class of empty graphs. Thus, we obtain as an application of our method the resultthat tree-depth is FPT, a result previously known from other algorithmic meta theorems(see [133, Theorem 17.2]).
The method of establishing that a parameterized problem is FPT by establishing thatit is slicewise nowhere dense and slicewise first-order definable appears to be a powerfulmethod of some generality which will find application beyond these examples.
Our work builds on a recent result of Grohe et al. [92] which shows that the problemof evaluating first-order formulas on any nowhere dense class of graphs is FPT with thesize of the formula as parameter. We extract from their proof of this result a generalstatement about the fixed-parameter tractability of definable sparse graph classes. Ourmethod is an adaptation of the main algorithm in [92]. Since the proof is essentially amodification of their central construction we state the main results they prove and explainhow the proofs can be adapted for our purposes.
Section 8.1 gives an overview of the key elements of the construction from [92] andthe elements from it which we need to extract for our result. Section 8.2 then gives ourmain result and Section 8.3 derives some consequences for distance measures.
8.1 Evaluating formulas on nowhere dense classes
The key result of [92] is:
Theorem 8.1.1 (Grohe et al. [92], Theorem 1.1). For every nowhere dense class C andevery ε > 0, every property of graphs definable in first-order logic can be decided in timeO(n1+ε) on C.
We first give a sketch of the algorithm from Theorem 8.1.1 with an emphasis on thechanges needed for our purposes. We refer to [92] for several definitions and results.
The algorithm developed in the proof of Theorem 8.1.1 uses a locality-based approach,similar to that used by Frick and Grohe [75] to show that first-order evaluation is FPT ongraphs of local bounded tree-width and developed in [50] for application to graph classeswith locally excluded minors. The idea is that any first-order formula ϕ is, by Gaifman’stheorem, equivalent to a Boolean combination of local formulae, that is formulae thatassert the existence of neighbourhoods satisfying certain conditions. In classes of sparsegraphs where the size (or other parameter) of neighbourhoods of a given radius can bebounded, this yields an efficient evaluation algorithm.
In nowhere dense classes of graphs, we cannot in general bound the size of neighbour-hoods. For example, the class of apex graphs is nowhere dense, but a graph may contain
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a vertex whose neighbourhood is the whole graph. However, nowhere dense classes arequasi-wide, which means that we can remove a small (i.e. parameter-dependent) set of ver-tices (the bottleneck) to ensure that there are many vertices that are far away from eachother. This is a notion of sparseness for graphs introduced in [50] (cf. Definition 2.3.25and Definition 2.3.26; for the connection to nowhere-dense classes, see [133]). Grohe etal. [92] use this approach to iteratively transform the input graph into a coloured graphwhere key bottleneck vertices are removed and vertices are coloured to keep relevant in-formation. At the same time the formula ϕ to be evaluated is also transformed so thatit can be evaluated on the modified graph. This procedure terminates on nowhere denseclasses of graphs within a constant number of steps.
A key data structure used in the algorithm is a neighbourhood cover, i.e. a collection ofconnected subgraphs, called clusters, so that each neighbourhood of a vertex is containedin one of the clusters. The radius of a cover is the maximum radius of any of its clusters.The degree of a vertex in a cover is the number of clusters the vertex is contained in.An important result from [92] is that graphs from a nowhere dense class allow for smallcovers and that such a cover can be efficiently computed.
Theorem 8.1.2 (Grohe et al. [92], Theorem 6.2). Let C be a nowhere dense class ofgraphs. There is a function f such that for all r ∈ N and ε > 0 and all graphs G ∈ Cwith n ≥ f(r, ε) vertices, there exists an r-neighbourhood cover of radius at most 2rand maximum degree at most nε and this cover can be computed in time f(r, ε) · n1+ε.Furthermore, if C is effectively nowhere dense, then f is computable.
In this theorem, f is a function of r and ε and depends on the class C in the sensethat it is determined, for an effectively nowhere dense C by its parameter function (seeDefinition 8.2.1 for a definition of parameter function). To be precise, the algorithm needsto order the vertices of G to witness a weak colouring number of less than nε. The weakcolouring number is an invariant of the graph that is guaranteed to be low for graphsfrom a nowhere dense class. The time bound f(r, ε) · n1+ε is obtained using an algorithmfor this from Nesetril and Ossona de Mendez [132].
While the algorithm of [92] assumes that the input graph G comes from the classC, we can say something more. For a fixed nowhere dense class C, where we know theparameter function h, we can, given G, r and ε, compute a bound on the running time ofthe algorithm from Theorem 8.1.2. By running the algorithm to this bound, we have thefollowing as a direct consequence of the proof of Theorem 8.1.2.
Lemma 8.1.3. There is a computable function f and an algorithm A which given anygraph G with n vertices and for any r ∈ N and ε > 0 either computes an r-neighbourhoodcover of radius at most 2r and maximum degree at most nε or determines that G 6∈ C.
At the core of the proof of Theorem 8.1.1 is the Rank-Preserving Locality Theorem.Given a neighbourhood cover X in a graph G, the algorithm iteratively removes bottleneckvertices and adds colours to the neighbourhoods of removed vertices to obtain a colouredgraph denoted G ?q+1
X q. Here q is an integer parameter obtained from the first-orderformula ϕ that we wish to evaluate in G. At the same time, ϕ is transformed into aformula ϕ that is (a) in the expanded signature of the G ?q+1
X q; and (b) in a logic FO+
which enriches FO by allowing us to assert distances between vertices without the needfor quantifiers. This ensures that the local sentence ϕ has the same quantifier rank asϕ, giving us the Rank-Preserving Locality Theorem below. In the following statement,
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a (q + 1, r)-independence sentence is a formula asserting the existence of a distance-r-independent set of size q + 1 of a particular colour.
Theorem 8.1.4 (Rank-Preserving Locality Theorem, Grohe et al. [92], Theorem 7.5).For every q ∈ N there is an r such that for every FO-formula ϕ(x) of quantifier rank qthere is an FO+-formula ϕ(x), which is a Boolean combination of (q + 1, r)-independencesentences and atomic formulas, such that for any graph G every r-neighbourhood cover Xof G, and every v ∈ V (G),
G |= ϕ(v) ⇐⇒ G ?q+1X q |= ϕ(v).
Furthermore, ϕ is computable from ϕ, and r is computable from q.
An important tool for constructing G ?q+1X q is a game characterisation of nowhere
dense classes. The game has three parameters: `,m, r. In the (`,m, r) Splitter game twoplayers Connector and Splitter play against each other. In each round Connector choosesa vertex u, and Splitter has to respond with a set A of vertices of size at most m in ther-neighbourhood of u. In the next round the graph is the neighbourhood of u with thevertices from A removed. If the graph is empty, Splitter wins. If Connector survives formore than ` rounds, she wins. Grohe et al. [92, Theorem 4.2] prove that if C is a nowheredense class, then there are `,m such that Splitter has a winning strategy on the (`,m, 2r)Splitter game on every graph in C.
The Splitter’s strategy on a graphG (which can be efficiently computed) is the essentialtool in the construction of G ?q+1
X q. The inductive procedure used to compute G ?q+1X q
from G is outlined in [92, Proof of Theorem 8.1]. We note that the termination of thealgorithm depends on the length of the game – which is bounded by a constant since C isnowhere dense. The strategy to compute Splitter’s moves is described in [92, Remark 4.3].Since the run time of the algorithm to compute G?q+1
X q only depends on q and the lengthof the Splitter game and we can compute this in advance, we can once again extract thefact that if we start with an arbitrary graph G, we can efficiently either transform it intoG ?q+1
X q or determine that it is not in the class C. This is summed up in the followinglemma.
Lemma 8.1.5. Let C be a nowhere dense class of graphs. For every ε > 0 there is analgorithm that runs in time O(f(q) · n1+ε) for some function f , and which given a graphG returns G ?q+1
X q or determines that G 6∈ C.
Theorem 8.1.4 reduces the problem of evaluating a formula of first-order logic todeciding a series of distance-r-independent set problems. So, the final ingredient is toshow that this is tractable. Formally, the problem is defined as follows:
Distance Independent SetInput: A graph G and k, r ∈ N.Parameter: k + rProblem: Does G contain an r-independent set of size k?
The problem is shown to be FPT on nowhere dense classes of graphs [92, Theorem 5.1].Since the runtime of the algorithm depends on the length of the Splitter game and Split-ter’s strategy, and this can be bounded in advance, [92, Theorem 5.1] can be restated asfollows:
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Lemma 8.1.6. Let C be a nowhere dense class of graphs. Then there is an algorithm Aand a computable function f such that for every ε > 0 A runs in time f(ε, r, k) · |V (G)|1+εand either solves the Distance Independent Set problem or determines that G 6∈ C.
This is all we need to evaluate ϕ on G ?q+1X q, which is equivalent to evaluating ϕ on
G by Theorem 8.1.4.
8.2 Deciding first-order definable nowhere dense prob-
lems
The main result of [92] establishes that checking whether G |= ϕ is FPT when parameter-ized by ϕ provided that G comes from a known nowhere dense class C. Thus, the formulais arbitrary, but the graphs come from a restricted class. In Section 8.1 above we give anaccount of this proof from which we can extract the observation that the algorithm can bemodified to work for an arbitrary input graph G with the requirement that the algorithmmay simply reject the input if G is not in C. This suggests a tractable way of decidingG |= ϕ provided that ϕ defines a nowhere dense class. Now the graph is arbitrary, but theformula comes from a restricted class. We formalise the result in the following theorem:
We use the model theory based notions of parameterized complexity introduced inSection 2.2.4 below. Additionally we define the notions of slicewise nowhere dense andslicewise first-order definable. The idea of slicewise definability of problems in a logic wasintroduced by Flum and Grohe [71].
We first introduce the notion of slicewise nowhere dense. The idea is that each slice isnowhere dense, i.e. for each value k of the parameter, the subclass of problem instanceswith parameter value bounded above by k is nowhere dense.
Definition 8.2.1. Let σ be a graph signature. We say that a parameterized graphproblem (Q, κ) is slicewise nowhere dense if there is a computable function h from pairsof integers to graphs such that for all i ∈ N, we have if G ∈ Q and κ(G) ≤ i then for allr we have h(i, r) 6r G. We call h the parameter function of Q.
The idea of slicewise definability is formalised in a similar way: a problem is slicewisefirst-order definable, if, for each value k of the parameter, there is a first-order formulathat defines the problem instances bounded above by k.
Definition 8.2.2 (Flum and Grohe [71]). Let σ be a signature. A parameterized problem(Q, κ) is slicewise first-order definable if there is a computable function ϕ : N → FO[σ]such that a σ-structure A with κ(A) ≤ i is in Q if, and only if, A |= ϕ(i).
Given these two definitions we can state the main theorem of this chapter:
Theorem 8.2.3. Let (Q, κ) be a problem that is slicewise first-order definable and slice-wise nowhere dense. Then (Q, κ) is fixed-parameter tractable.
Proof. In the following, for ease of exposition, we assume that an instance of the problemconsists of a graph G and κ(G) = i for some positive integer i.
Step 1: Compute ϕ and the parameter function. Since (Q, κ) is slicewise first-orderdefinable, we can compute from i a first-order formula ϕ = ϕ(i) which defines the
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class of graphs Ci = H | H ∈ Q and κ(H) ≤ i. Moreover, since (Q, κ) is slicewisenowhere dense, we can compute from i an algorithm that computes the parameterfunction h for Ci.
Step 2: Obtain ϕ from ϕ. By the Rank-Preserving Locality Theorem (Theorem 8.1.4),we can compute from ϕ the formula ϕ and a radius r.
Step 3: Find a small cover X for G. By Lemma 8.1.3, we can either find a cover Xfor G, or reject if the algorithm determines that G 6∈ Ci.
Step 4: Simulate Splitter game to compute G ?q+1X q. By Lemma 8.1.5 we obtain
G ?q+1X q or reject if the algorithm determines that G 6∈ Ci.
Step 5: Evaluate ϕ on G ?q+1X q. Finally we evaluate ϕ on G?q+1
X q. To do this, we needto solve the distance independent set problem. We can do this by Lemma 8.1.6.
Since evaluating ϕ on G ?q+1X q is equivalent to evaluating ϕ on G this allows us to
decide whether G ∈ Q.
8.3 Applications
In this Section we discuss some applications of Theorem 8.2.3 that demonstrate its power.We begin by considering simple edit distances.
8.3.1 Deletion distances
Recall from Definition 3.1.1 that a graph G has deletion distance k to a class C if thereexists a set S of k vertices in G so that G \ S ∈ C. Suppose (Q, κ) is a parameterizedgraph problem. We define the problem of deletion distance to Q as follows:
Deletion Distance to QInput: A graph G and k, d ∈ N.Parameter: k + dProblem: Does G contain a set S of k vertices so that κ(G\S) ≤ d and G\S ∈ Q?
In many of the examples below, we define formulas of first-order logic by relativisation.For convenience, we define the notion here.
Definition 8.3.1. Let ϕ and ψ(x) be first-order formulas, where ψ has a distinguishedfree variable x. The relativisation of ϕ by ψ(x), denoted ϕ[x.ψ] is the formula obtainedfrom ϕ by replacing all subformulas of the form ∃v ϕ′ in ϕ by ∃v(ψ[v/x] ∧ ϕ′), and allsubformulas of the form ∀v ϕ′ in ϕ by ∀v(ψ[v/x] → ϕ′). Here ψ[v/x] denotes the resultof replacing the free occurrences of x in ψ with v in a suitable way avoiding capture.
The key idea here is that ϕ[x.ψ] is true in a graph G if, and only if, ϕ is true in thesubgraph of G induced by the vertices that satisfy ψ(x). Note that the variable x thatis free is ψ is bound in ϕ[x.ψ]. Other variables that appear free in ψ remain free in ϕ[x.ψ].We stress this as it is needed in Proposition 8.3.5 where nested relativisations are used.
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Proposition 8.3.2. If (Q, κ) is slicewise nowhere dense and slicewise first-order definablethen Deletion Distance to Q is FPT.
Proof. It suffices to show that Deletion Distance to Q is also slicewise nowhere denseand slicewise first-order definable. For the latter, note that if ϕi is the first-order formulathat defines the class of graphs Ci = G | κ(G) ≤ i and G ∈ Q, then the class of graphsat deletion distance k to Ci is given by:
∃w1, . . . , wkϕ[x.θk]i
where θk(x) is the formula∧
1≤i≤k x 6= wi.
To see that Deletion Distance to Q is also slicewise nowhere dense, let h be theparameter function for Q. If the graph h(i, r) has m vertices, then Km is not a depth-r-minor of any graph in Ci. Then a graph which has deletion distance k to Ci cannothave Km+k as a depth-r-minor. Indeed, suppose Km+k r G and G \ S ∈ Ci where S isa set of k vertices. Vertices from S can appear in the images of at most k vertices fromKm+k under the minor map. Thus, this minor map also witnesses that Km r G \ S, acontradiction.
Consider as an example deletion distance k to maximum degree d.
Example 8.3.3. We can express that a vertex has no more than d different neighboursusing the following formula:
ϕd(v) := ¬∃v1, . . . , vd+1
((∧i<j
vi 6= vj
)∧
(∧i
E(v, vi)
)).
So a graph G has maximum degree bounded by d if, and only if, G |= ∀vϕd(v). Let Cdbe the class of graphs with maximum degree bounded by d. Then the following formulacaptures deletion distance k to maximum degree d:
∃w1, . . . , wk∀v
(∧i
v 6= wi
)→(
¬∃v1, . . . , vd+1
((∧i,j
vi 6= wj
)∧
(∧i<j
vi 6= vj
)
∧
(∧i
E(v, vi)
))).
Thus the technique introduced in this section is sufficient to show that deletion distancek to maximum degree d is fixed-parameter tractable parameterized by k + d.
Moser and Thilikos [129] showed that deleting k vertices to obtain a d-regular graphis fixed-parameter tractable parameterized by k+d. Since the class of d-regular graphs isalso first-order definable and nowhere dense for any d, their result is also a consequenceof Theorem 8.2.3.
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8.3.2 Edit distances to graph classes defined by degree con-straints
Instead of deletion distance (defined by deleting vertices), we can also consider moregeneral graph editing distances (defined through more general edit operations on thegraph), e.g. modifying the graph by adding or deleting edges. (See Definition 3.3.1 for aformal definition of edit distance.)
These more general editing operations are also first-order definable. For example, toobtain a formula that defines the graphs that are one edge addition away from a class ofgraphs C defined by the formula ϕ, we construct ϕ from ϕ by replacing all occurrences ofE(w1, w2) in ϕ by:
(w1 = u ∧ w2 = v) ∨ (w1 = v ∧ w2 = u) ∨ E(w1, w2).
Then the formula ∃u∃vϕ defines the class of graphs with edge addition distance 1 to C,i.e. G |= ∃u∃vϕ if, and only if, G with an additional edge satisfies ϕ. This can easilybe generalised to k edge additions: From a formula ϕ we can obtain a formula ϕk suchthat for any graph G we have that G |= ϕ if, and only if, there are pairs of verticesu1, v1, . . . , uk, vk ∈ V (G) such that G, with additional edges u1v1, . . . ukvk, satisfies ϕ.
Similarly, given a formula ψ that defines a graph class C, we can define a formula ψby replacing all occurrences of E(w1, w2) in ψ by:
(w1 6= u ∨ w2 6= v) ∧ (w1 6= v ∨ w2 6= u) ∧ E(w1, w2).
Then ∃u∃vψ defines the class of graphs with edge deletion distance 1 to C. It should beclear that this can also be generalised to k edge deletions.
Thus, an analogue of Proposition 8.3.2 can be obtained for any edit distance wherethe allowed edit operations are a combination of vertex and edge deletions and additions.In the following we discuss this in more detail, where the class we are editing to is definedby degree constraints.
In his doctoral thesis Mathieson [123] studies the parameterized complexity of suchgraph editing problems with the aim of satisfying a variety of degree constraints. Hedefines the general template of Weighted Degree Constrained Editing (or justWDCE). In the following we explore one instance (that he refers to as WDCEr
1) of anumber of more general templates that also allow for weight functions of vertices andedges, as well as a different degree target for each vertex and a target for the sum of edgeweights. This is just for the sake of simplicity here, the more general operations are alsodefinable in first-order logic and give rise to nowhere dense graph classes. In the followingwe abbreviate the editing operations vertex deletion, edge deletion and edge addition asv, e and a respectively. Then for each non-empty S ⊆ v, e, a define WDCE(S) as:
Weighted Degree Constrained Editing(S) (WDCE(S))Input: A graph G, two integers k and dParameter: k + dProblem: Can we obtain from G a graph G′ using at most k editing operationsfrom S only, such that all vertices of G′ have degree d?
The problem is W[1]-hard parameterized by k [89, 123], so we need to consider a jointparameterization. Mathieson [125] shows that the problem is fixed-parameter tractable
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for any S and parameter k + d. Inspired by Stewart [149], Mathieson shows that theproblem is first-order definable (with the size of the formula depending on k and d), byconsidering the incidence graph as a relational structure. (For the weighted version of theproblem, he adds a unary relation for every possible weight.) Note that a graph G thatcan be edited to be regular in k steps can have maximal degree at most k + d and G istherefore also nowhere dense. Thus Mathieson’s fixed-parameter tractability results alsofollow directly from Theorem 8.2.3.
Building on this, Golovach [88] gives a concrete algorithm that edits a graph so thatevery vertex has a given degree at most d using at most k edge additions/deletions.
More recently, Mathieson [124] looks at more general versions of degree constraintproblems. He considers three notions of regularity: edge-degree-regular, edge-regular andstrongly-regular. He studies the problems of editing to these three notions of regularity.
The edge-degree of an edge uv is the sum of the degrees of the endpoints of d(u)+d(v)and a graph is edge-degree-regular if all edges uv have the same edge-degree.
The two other notions combine the degrees of vertices and common neighbourhoodsof endpoints of edges (and non-edges). A graph is (r, λ)-edge-regular if every vertexhas degree r and every edge uv has |N(u) ∩ N(v)| = λ. A graph is (r, λ, µ)-strongly-regular if it is (r, λ)-edge-regular and for every pair u, v of non-adjacent vertices we have|N(u)∩N(v)| = µ. This is the standard notion of a strongly regular graph as introducedby Bose [22].
Just as above we abbreviate the editing operations vertex deletion, edge deletion andedge addition as v, e and a respectively. We also abbreviate the three notions of regularityedge-degree regular, edge-regular and strongly regular as r1, r2, r3 respectively. Then foreach non-empty S ⊆ v, e, a and each r ∈ r1, r2, r3 define RCE(S, r) as:
Regularity Constrained Editing(S, r) (RCE(S, r))Input: A graph G, integers k, d (and additionally an integer λ when r = r2, and apair of integers (λ, µ) when r = r3)Parameter: k + dProblem: Can we obtain from G a graph G′ using editing operations from S only,such that G′ is r-regular (with the given parameters)?
Mathieson [124] shows that editing to these three notions of regularity is FPT param-eterized by k + d, where d is the degree we are editing to and k is the number of allowededits. This also follows from our meta-theorem: Just as in Example 8.3.3, it is easy toshow that the class of graphs with edit distance k to an d-regular graph is first-orderdefinable and has bounded degree, and is thus also nowhere dense. The additional con-straints are also first-order definable, for example |N(u) ∩N(v)| = λ can be expressed asfollows:
∃x1 . . . ∃xλ.
(∧i<j
xi 6= xj∧i
(E(xi, u) ∧ E(xi, v))
∧¬∃y
(∧i
(xi 6= y) ∧ (E(u, y) ∧ E(v, y))
))So each of these problems is first-order definable and nowhere-dense, so it follows directlyfrom Theorem 8.2.3 that these problems are fixed-parameter tractable with the combinedparameter k + d.
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8.3.3 Tree-depth
Recall that tree-depth (cf. Chapter 4) is a graph parameter that lies between the widelystudied parameters vertex cover number and tree-width. It has interesting connectionsto nowhere dense graph classes, and can itself be interpreted as a distance measure (Weshowed that elimination distance, introduced in Chapter 5, can be thought of as a naturalgeneralisation of tree-depth: tree-depth is just the elimination distance to the class ofedgeless graphs. A graph has tree-depth k if, and only if, it has elimination distance k tothe class of empty graphs; cf. Section 5.4). For convenience we repeat the usual definition(Definition 4.0.4) here:
Definition 8.3.4. The tree-depth of a graph G, written td(G), is defined as follows:
td(G) :=
0, if V (G) = ∅;1 + mintd(G \ v) | v ∈ V (G), if G is connected;
maxtd(H) | H a component of G, otherwise.
It is known that the problem of determining the tree-depth of a graph is FPT, withtree-depth as the parameter (see [133, Theorem 7.2]). We now give an alternative proofof this, using Theorem 8.2.3. First we show below that the class of graphs of tree-depthat most k is first-order definable. This is an interesting fact in itself and we are not awareif this has been established elsewhere.
Proposition 8.3.5. For each k ∈ N there is a first-order formula ϕk such that a graphG has tree-depth k if, and only if, G |= ϕk.
Proof. We use Lemma 4.1.1 which states the fact that in a graph of tree-depth less thank, there are no paths of length greater than 2k. This allows us, in the inductive definitionof tree-depth above, to replace the condition of connectedness (which is not first-orderdefinable) with a first-order definable condition on vertices at distance at most 2k.
Let distd(u, v) denote the first-order formula with free variables u and v that is satisfiedby a pair of vertices in a graph G if, and only if, they have distance at most d in G. Notethat the formula dist
[x.x 6=w]d (u, v) is then a formula with three free variables u, v, w which
defines those u, v which have a path of length d in the graph obtained by deleting thevertex w.
We can now define the formula ϕk by induction. Only the empty graph has tree-depth0, so ϕ0 := ¬∃v(v = v).
Suppose that ϕk defines the graphs of tree-depth at most k, let
θk := (∀u, v dist2k+1(u, v)) ∧ ∃w(ϕ[x.x 6=w]k ).
The formula θk defines the connected graphs of tree-depth at most k+1. Indeed, the firstconjunct ensures that the graph is connected as no pair of vertices has distance greaterthan 2k+1 and the second conjunct ensures we can find a vertex w whose removal yieldsa graph of tree-depth at most k.
We can now define the formula ϕk+1 as follows.
ϕk+1 := (∀u, v dist2k+1+1(u, v)→ dist2k+1(u, v)) ∧ ∀wθ[x.dist2k+1 (w,x)]
k .
The formula asserts that there are no pairs of vertices whose distance is strictly greaterthan 2k+1 and that for every vertex w, the formula θk holds in its connected component,namely those vertices which are at distance at most 2k+1 from w.
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It is well known that for any k, the class of graphs of tree-depth at most k is nowheredense. It follows for example from the characterisation of nowhere dense classes in The-orem 2.3.33 via low tree-depth colourings (cf. Definition 2.3.32) that the class of graphsof tree-depth at most k has bounded expansion and is nowhere dense. Now, using The-orem 8.2.3, this fact together with Proposition 8.3.5 gives an alternative proof of thefollowing:
Proposition 8.3.6. The problem of determining, given a graph G and an integer k,whether G has tree-depth at most k is fixed-parameter tractable parameterized by k.
While the proof of Proposition 8.3.2 shows that deletion distance to any slicewisefirst-order definable class is also slicewise first-order definable, Proposition 8.3.5 showsthat elimination distance to the particular class of empty graphs is slicewise first-orderdefinable. It does not establish this more generally for elimination distance to any slicewisenowhere dense class – that remains an open question. We conjecture that there is aslicewise nowhere dense class C for which elimination distance to C is not first-orderdefinable.
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CHAPTER 9
Distance parameters to sparsegraph classes: two case studies
In this chapter we study the parameterized complexity of two more natural distance pa-rameters, i.e. graph problems where the natural parameterization is a distance parameter.
First we consider the Clique Dominating Set problem on nowhere dense classesof graphs that are closed under induced subgraphs. The problem takes a graph G andnumbers k, ` as input and asks whether k vertices are sufficient to dominate all the cliquesof G of size `. We show that Clique Dominating Set is FPT on nowhere dense classesof graphs, parameterized by k. The size of the smallest set that dominates all `-cliques ofa graph can be seen as a relaxed deletion distance to the class of graphs excluding K` asa subgraph.
The fixed-parameter tractability of Clique Dominating Set follows from a recentalgorithmic meta-theorem of Grohe et al. [92]. We establish the result in a different wayand believe our result is of independent interest.
The other problem we study in this chapter is the Anchored k-core problem. Recallfrom Definition 3.7.3 that the k-core of a graph G is the maximal subgraph of G in whichall vertices have at least k neighbours. A b-anchored k-core (see Definition 9.2.1 below)relaxes the notion of the k-core and allows an additional number of b vertices in the corethat may have degree less than k. The Anchored k-core problem asks, given a graphG, and integers k, b, p ∈ N, whether G has a b-anchored k-core of size at least p. The sizeb of the ‘anchor’ can be seen as a distance measure to the the class of graphs that have ak-core of at least a certain size p.
The Anchored k-core problem is known to be W[2]-hard with respect to this dis-tance parameter b, called the budget, so we consider joint parameterizations with anotherparameter: the tree-width of G. We show that the problem is FPT parameterized by thetree-width tw(G) and budget b.
We also observe that the Anchored k-core problem can be expressed in monadicsecond-order logic (MSO), where the size of the MSO formula is bounded by a functionof k and p. Using a generalisation of Courcelle’s Theorem (Theorem 3.5.5) this impliesthat the Anchored k-core problem is also fixed-parameter tractable parameterized bythe clique-width (Definition 3.5.2) of G and additional parameters k and p.
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9.1 Clique Dominating Set
In this section we establish that Clique Dominating Set is FPT on nowhere-densegraph classes that are closed under induced subgraphs. The problem asks, given a graphG and two non-negative integers k and `, whether there is a set of k vertices that domi-nates all the cliques of size `. The parameterization we consider is the size of the cliquedominating set k.
First we introduce some notation:
Definition 9.1.1. Let G be a graph. We say that a set of vertices S ⊆ V (G) dominatesa set of vertices T ⊆ V (G) (or, that S is a dominating set for T ) if there is a vertex s ∈ Sand a vertex t ∈ T such that either s = t or st ∈ E(G).
Definition 9.1.2. Let G be a graph. An `-clique dominating set for G is a set S thatdominates all cliques of cardinality ` in G. The `-clique domination number of G is thesize of a minimal `-clique dominating set for G.
The notion of `-clique domination number is related to that of deletion distance (seeDefinition 3.1.1) in the following sense: Clearly, the deletion distance to the class of graphsexcluding K` as a subgraph is an upper bound for the `-clique domination number: ifafter deleting k vertices from the graph there are no cliques of size ` left, then the `-clique domination number is at most k, because a subset of the k deleted vertices mustbe an `-clique dominating set. However, the converse is not true: if we have an `-cliquedominating set of size k, then removing that set is not sufficient to remove all `-cliques.As a simple example, consider a disjoint union of m copies of an `-clique, all connectedto a single additional apex vertex a. Of course this vertex a gives us a dominating set ofsize 1, but the deletion distance to the class of graphs excluding K` is m.
Thus the `-clique domination number can be interpreted as a relaxed deletion dis-tance to the class of graphs excluding K` as a subgraph. Given a graph G with `-cliquedomination number k we can obtain a graph that does not contain K` as a subgraph byremoving all the vertices in a minimal clique dominating set, and all the neighbours ofthese vertices. The class of graphs excluding K` as a subgraph is a sparse graph class.
Formally the Clique Dominating Set on a nowhere dense class C is defined asfollows:
Clique Dominating SetInput: A graph G ∈ C, and non-negative integers k, `Parameter: kProblem: Does G have `-clique domination number at most k, i.e. is there a setof vertices D ⊆ V (G) with |D| ≤ k such that every clique in G of size ` contains avertex that is either in D or adjacent to a vertex in D?
The Clique Dominating Set problem was introduced by Flum and Grohe [72, p.100] as an example of an A[2]-complete problem, which implies that it is W[2]-hard. Givenits complexity status, it is unlikely to be FPT in general. That is why we are consideringit here restricted to nowhere dense graph classes.
Flum and Grohe consider a version of the problem that is parameterized by k + `.However, we are considering the problem for graphs from a nowhere dense class C, andthus there is an integer `′ such that K`′ is not a subgraph of any graph in C. So if ` > `′ the
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problems becomes trivial – the `-clique dominating number is 0 for all G inC. Otherwise` is bounded by a constant. For that reason we consider a parameterization by just k.
The problem is first-order definable; it is expressed by the following formula:
∃d1 . . . ∃dk.∀c1, . . . , c`.
( ∧1≤i<j≤`
E(ci, cj)
)→( ∨
1≤i≤k,1≤j≤`
di = cj ∨ E(di, cj)
)
A recent meta-theorem of Grohe et al. [92] establishes that first-order model checkingis FPT on nowhere dense classes of graphs (parameterized by the size of the formula).So that result, together with the slicewise first-order definability of the problem, alreadyimplies that Clique Dominating Set is FPT on nowhere dense classes of graphs.
In this section we prove that Clique Dominating Set is FPT in a different way. Ourapproach is problem-specific, and uses an alternative characterisation of nowhere densegraph classes, and a powerful algorithmic result by Dawar and Kreutzer [52]. We thusbelieve our result is of independent interest.
In the following we use an alternative characterisation of nowhere dense graph classescalled quasi-wideness (see Definition 2.3.26). Nesetril and Ossona de Mendez [133] haveshown that a class of graphs that is closed under taking induced subgraphs is nowheredense if and only if it is quasi-wide (cf. Theorem 2.3.27).
Dawar and Kreutzer [52] extracted the algorithmic content of that theorem as a toolto prove that the Distance-d Dominating Set problem is fixed-parameter tractableon classes of graphs that are nowhere dense and closed under induced subgraphs; theyproved the following:
Lemma 9.1.3 ([52, Lemma 9]). Let C be a nowhere dense class of graphs and let h bethe parameter function for C, i.e. a function such that Kh(r) 6r G for all G ∈ C. Thenthe following problem is solvable in time O(|G|2):
Compute Quasi-Wide ParametersInput: G ∈ C, r,m ∈ N,W ⊆ V (G) with |W | > N(h(r), r,m)Problem: Compute a set S ⊆ V (G), |S| ≤ h(r)− 2 and a set A ⊆ W with |A| ≥ msuch that A is r-scattered in G− S.
Using this lemma we now prove that Clique Dominating Set problem is FPT onclasses of graphs that are nowhere dense and closed under induced subgraphs.
Theorem 9.1.4. The Clique Dominating Set problem is FPT on classes of nowheredense graphs that are closed under induced subgraphs.
Proof. Let C be a class of nowhere dense graphs and let G ∈ C. We show that we canreduce the problem to a bounded number of subproblems with parameter k − 1.
Recall that since C is a nowhere dense class we can assume that ` is bounded by aconstant. Thus we first can efficiently test whether a vertex is either contained in an`-clique (i.e. its neighbourhood contains an (`− 1)-clique) or adjacent to an `-clique (i.e.one of its neighbours is contained in an `-clique ). We remove all the vertices that are
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neither contained in an `-clique nor adjacent to one, obtaining an induced subgraph G′
of G. Since C is closed under taking induced subgraphs, we have that G′ ∈ C.Next we use Lemma 9.1.3. If |V (G′)| < N(h(3), 3, k + 1), we just find a solution by
brute force. Otherwise we obtain a subset S ⊆ V (G′) and a subset A ⊆ V (G′) with|A| > k + 1 such that A is 3-scattered in G′ \ S. Now every vertex v ∈ A contains a
clique in its neighbourhood NG′\S2 (v), but for any w ∈ A, w 6= v, we have N
G′\S3 (v)(v) ∩
NG′\S3 (v)(w) = ∅. So if a vertex dominates one of the cliques, it cannot dominate any of
the others. Thus G′ \ S does not contain an `-clique dominating set of size k. So if thereis an `-clique dominating set for G′ of size k it must contain a vertex from S.
Now for each s ∈ S, we remove s and N(s) from G′ to obtain the induced subgraphG′s. We have that G has an `-clique dominating set of size k if and only if one of the G′shas an `-clique dominating set of size k − 1. Moreover, for each s ∈ S we have G′s ∈ C.
The size of S is bounded by the constant h(3), and the number of reduction stepsis bounded by k. Moreover, we can find an `-clique in time |G|`. So the overall time isbounded by h(3)k · (|G|2 · |G|` + |G|2).
Note that that this argument can be generalised to domination problems for otherstructures in a graph than `-cliques. In fact, any problem variation where the structurecan be efficiently tested for is FPT. For example the problem whether there is a setof k vertices that dominates all independent sets of size ` is fixed-parameter tractableparameterized by k + `.
9.2 The Anchored k-core problem
In this section we explore two different parameterizations of the Anchored k-coreproblem. Recall from Definition 3.7.3 that the k-core of a graph G is the largest inducedsubgraph H of G such that every vertex in H has at least k neighbours. The notion isclosely related to that of the degeneracy of a graph: if G has a non-empty k-core, then Ghas degeneracy at least k, and, conversely, the degeneracy of G is the largest k for whichG has a non-empty k-core.
The notion of k-core is important for applications. The k-core of a graph has beenused in the context of social networks for some time now [148], where it is being usedas a measure for the cohesion of the network. More recently, in the context of socialnetworking websites, it was shown that the k-core of a social network graph is mostlycongruent with the active user base: users contribute more content if at least a certainnumber of their friends do [31].
In the same context Bhawalkar et al. [16] used the idea of the k-core to suggest a simplemodel that measures how healthy a social networking website is: if a user is engaged withthe site only if a sufficient number of friends, say k friends, are, then we can observe aprocess of ‘unravelling’ until only the k-core is left. This raises an interesting question:can the size of the k-core be increased by influencing the engagement of users? If it ispossible to convince a certain amount of users to stay, even if their number of friends isbelow the threshold, can the ‘unravelling’ be stopped much earlier?
This is exactly the Anchored k-core problem, which was recently introduced byBhawalkar et al. [16]. We are given a budget b, a degree threshold k and a target size pand the question is whether it is sufficient to convince b users to stay, so that after the
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process of unravelling stops, where a user leaves if they have fewer than k friends, thereare still p users left.
To formalize the idea of fixing vertices, we introduce the notion of an anchored k-core:
Definition 9.2.1. Let G be a graph and let b, k ≥ 0. A b-anchored k-core of G is asubgraph H of G such that at most b vertices of H have degree less than k. We refer tothe set of vertices of H with degree less than k as anchor set.
Note that the size b of the ‘anchor’ gives us an interesting notion of distance. Itmeasures how far a graph is from unravelling: Instead of deleting vertices, we ask howmany vertices have to be ‘fixed’ in order to guarantee a k-core of size at least p. This alsogives us a lower bound for the graph editing problem of how many vertices we need toadd in order to guarantee a k-core of size at least p.
Formally, the Anchored k-core problem is defined as follows.
Anchored k-coreInput: A graph G, positive integers k, b, p ∈ NProblem: Does G contain a b-anchored k-core of size at least p, i.e. is there asubgraph H of G of size |H| ≥ p such that at most b vertices in H have degree lessthan k?
Bhawalkar et al. [16] show that for k = 2, there is a polynomial-time algorithm.However, for k ≥ 3 the problem becomes much harder and does not even admit a poly-nomial time approximation algorithm: for all k ≥ 3, it is NP-hard to obtain an O(n1−ε)approximation for every positive constant ε > 0, i.e. they prove that it is NP-hard to dis-tinguish between instances of Anchored k-core instances where an optimal anchoredk-core contains Ω(n) vertices and where O(b) vertices in the anchored k-core suffice [16,Theorem 3].
So the problem is quite hard from the perspective of classical complexity theory. Thismotivates Bhawalkar et al. [16] to study the parameterized complexity of the problem.They consider the budget b as parameter, but it turns out that for every k ≥ 3, theproblem is W[2]-hard with respect to the parameter b.
The hardness of the problem parameterized by the budget motivates Bhawalkar etal. [16] to consider additional structural parameterisations. They successfully show thatthere is an algorithm that runs in time O((3(k + 1)2)w · b2 · n), where w is the tree-widthof the graph. The result is obtained using standard dynamic programming techniques onthe tree decomposition of the graph. This shows that the problem is FPT parameterizedby k and the tree-width of the input graph.
Chitnis et al. [36] establish further hardness results: they show Anchored k-core isW[1]-hard parameterized by p, even for k = 3. This improves the result of Bhawalkar etal. [16] since we can always assume that p ≥ b. They also show that the problem remainsNP-hard even on planar graphs for all k ≥ 3, and even if the maximum degree of thegraph is k + 2.
But what about parameterizing by the natural distance parameter b on restrictedgraph classes? Chitnis et al. [36] show that Anchored k-core is FPT on planar graphsparameterized by b for all k ≥ 7.
In our analysis we consider further joint parameterizations, combining the distanceparameter b with the structural distance parameters tree-width (Definition 3.4.2) andclique-width (Definition 3.5.2).
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In Section 9.2.1 we study the Anchored k-core problem parameterized by b andtree-width and show that it is fixed-parameter tractable.
In Section 9.2.2 we show that the Anchored k-core is MSO-definable, with the sizeof the formula depending on k and b. Using a generalisation of Courcelle’s Theorem toclique-width this implies that Anchored k-core is FPT parameterized by k, b and theclique-width of the graph.
Lastly we talk about some open questions in Section 9.2.3, in particular we discusshow the techniques in Section 9.2.1 can be used to extend the result to more generalparameterizations, such as the combination of b and the degeneracy of the graph.
9.2.1 Anchored k-core parameterized by budget and tree-width
In this section we show that the Anchored k-core problem is FPT parameterized bythe budget and tree-width of the graph. We do this by splitting the problem into twocases: we deal separately with the case where k is small and the case where k is large.
We generalise a lemma from Chitnis et al. [36] for the case where k is large. Chitnis etal. [36] observe that, for any p ≥ 0, it is fairly straightforward to write down a first-ordersentence that is true on graphs that have an anchored k-core of size at most p. (Thedefinition literally translates to FO.) The size of the first-order formula depends on b, kand p, but since b, k ≤ p, it is bounded by a function of p. So if we know that the graphmust contain a small k-core, the problem reduces to evaluating first-order logic formulas.
A closer look at the proof by Chitnis et al. [36] reveals that the argument is not specificto planar graphs. They use the well known degeneracy (Definition 3.7.1) of planar graphs,i.e. the property of planar graphs that the number of edges is bounded by 3|V | − 6, toshow that the ratio of high degree vertices to all vertices is bounded:
Lemma 9.2.2. Let G be a planar graph, n = |G|,m = |v ∈ G | deg(v) ≥ k|. Then, ifk ≥ 7, we have that m/n < 6
7.
Chitnis et al. [36] use the lemma to show that the Anchored k-core problem isFPT on planar graphs parameterized by b for all k ≥ 7. This is an easy application of thislemma: For sufficiently large k (k > 7), the above lemma implies that the k-core must besmall, and we can check it using a small first-order formula.
In the following we prove a more general version of Lemma 9.2.2. The property ofplanar graphs that is used in the proof is their degeneracy. So a similar statement canbe made for any graph of bounded degeneracy. It again follows that for sufficiently largek, the k-core must be small, so we solve the problem by evaluating a small first-orderformula. We can generalise the proof of Lemma 9.2.2 to graphs of bounded degeneracyas follows:
Lemma 9.2.3. Let G = (V,E) be a d-degenerate graph with no isolated vertices. Letn = |V (G)| and m = |v ∈ G | deg(v) ≥ k|. Then, if k > 2d, we have that m/n ≤ 1− 1
2d.
Proof. Since G is d-degenerate, we have that |E(G)| ≤ dn. So the following holds:
2dn ≥ 2|E(G)|
=∑
v∈V (G)
deg(v)
≥ (2d+ 1)m+ n−m.
Rearranging the terms gives m/n ≤ 1− 12d
.
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This is all we need to prove the main theorem of the section. The proof combines twotechniques: We apply Lemma 9.2.3 to solve the case where k is large. In that case weuse that the input graph is sparse and it therefore cannot contain a large k-core. Thuswe can reduce the problem to a model checking problem with a formula of bounded size.For small k, we can use the algorithm described by Bhawalkar et al. [16] for solving theproblem on graphs of bounded tree-width.
Theorem 9.2.4. The Anchored k-Core problem is FPT parameterized by the tree-width t of the input graph and the budget b.
Proof. We assume without loss of generality that the input graph G does not containisolated vertices (otherwise we just remove them).
We distinguish the cases where k is small and large. If k < 2t, we can use the algorithmin [16], which runs in time O((3(k + 1)2)t · b2 · n) = O(f(t)n3).
If k > 2t, it follows from Lemma 3.7.2 that because G has tree-width t it is alsot-degenerate. Assume that H is an optimal k-core with anchor set B ⊆ H. Since H is aninduced subgraph of G, it is also t-degenerate and we get the following using the abovelemma:
1− |B||H|
=|H| − |B||H|
=|H \B||H|
< 1− 1
2t.
Rearranging the terms gives us |H| < 2t|B| < 2tb. So we know that an anchored k-corehas size bounded by the parameter and express the problem in a first-order sentence ofsize bounded by a function of 2tb. Since G has bounded tree-width we can evaluate thesentence in FPT time.
In fact, the proof technique can be generalized to show that the problem is FPTparameterized by b on all t-sparse (t-degenerate) graphs for large enough values of thedegree threshold k, i.e. whenever k > 2t. However, we do not know how to deal withsmall values of k, and this part of the problem remains open.
9.2.2 Anchored k-core parameterized by degree, size of the coreand clique-width
Here we consider the Anchored k-core problem parameterized by the degree k, thesize of the core p and the clique-width of the input graph. We show that the problem isFPT with a joint parameterization of these three parameters. This is a consequence ofthe fact that the problem can be expressed in MSO with the size of the formula boundedby a function of k + p, and a generalisation of Courcelle’s theorem [42].
The main advantage that MSO gives us over first-order logic is that we can talk aboutsets without having to be specific about their size. So we can specify a lower bound forthe k-core and an upper bound for the budget: For a second-order variable X and aninteger x we can write a formula |X| ≥ x that is true if, and only if, the set X has atleast x elements, with the size of the formula bounded by a function of x. Similarly, fora second-order variable X and an integer x we can write a formula |X| ≤ x that holds if,and only if, the set X has at most x elements with the size of the formula bounded bya function x. It is also clear that for every non-negative integer k there is a first-orderformula δSk such that G |= δSk (v) if, and only if, degG[S](v) ≥ k.
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Also note that, if b ≥ p, the problem becomes trivial, because we can just fix any bvertices to obtain a large enough anchored k-core. So every graph of size at least p is apositive instance. Thus, in the following we assume without loss of generality that b < p.
To match second-order syntax, the problem can be rephrased as follows: Is there a setof vertices K of size at least p and a subset B of K of size at most b such that all verticesin K \B have degree in G[K] at most k? Or as a formula:
∃K.(|K| ≥ p ∧
(∃B.|B| ≤ b ∧ ∀v.
(K(v)∧ 6 B(v)→ δ
K\Bk (v)
)))So together with Theorem 3.5.5 this implies that the problem is FPT parameterized
by clique-width, k and p:
Theorem 9.2.5. The anchored k-core problem is FPT parameterized by the clique-widthc, k and p.
9.2.3 Open questions
An interesting question is whether the technique used for tree-width in Section 9.2.1can be pushed further. Our Lemma 9.2.3 is a general statement about graphs of boundeddegeneracy. We show that the problem is FPT parameterized by b and t on all t-degenerategraphs for large enough values of the degree k. Is there a way to deal with small valuesof k?
The most general result one could aim for with the techniques used here is to provethat Anchored k-core is FPT parameterized by degeneracy and budget. Bounded de-generacy is a very general notion of sparsity, and clases of bounded degeneracy includemany other graph classes, for example graphs of bounded expansion. We know the prob-lem for such classes is FPT for sufficiently large k. The open question is whether there isan FPT algorithm that can deal with the case where k is small – that would be enoughto show fixed-parameter tractability with parameters b and t.
Another interesting area for the problem is the study of the directed version. Sinceit was designed with application to social networks in mind, that would be very natural,as many such networks are better modelled using directed graphs. Chitnis et al. [35] alsodiscuss the directed version of the problem, and prove some hardness and tractabilityresults. It seems possible that some of the approaches discussed here might be helpful inthe direction version as well.
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CHAPTER 10
Further remarks and futureresearch directions
In this dissertation we introduced the term distance parameter to capture the idea of agraph parameter that measures how far the graph is from being contained in some class.We showed that many common graph parameteres can be interpreted in that way, andpresented several results to emphasise the usefulness of distance parameters.
Chapter 4 discussed the distance parameter tree-depth in some detail, and establishedfixed-parameter tractability and hardness results for a number of problems parameterizedby tree-depth.
In Chapter 5 we introduced a new distance parameter, elimination distance to a classC. The parameter measures distance in a way that generalises the notion of vertex deletionin a natural way by allowing deletions from every component in each deletion step. Weshow that the parameter tree-depth is a specific instance of elimination distance to aclass C, where C is the class of edgeless graphs. Indeed, elimination distance generalisesdeletion distance in the same way that tree-depth generalises vertex cover number.
We gave an application of the elimination distance in Chapter 6, where we demonstratethat Graph Isomorphism is FPT parameterized by elimination distance to boundeddegree, extending a result of Bouland et al. [23].
In Chapter 7 we also studied the parameter itself. We showed that given a finite set ofminors M that characterise a minor-closed class of graphs C, we can find the minors thatcharacterise the graphs with elimination distance k to C. This yields an explicit algorithmfor determining whether a graph has elimination distance k to a minor closed class C ifthe set M of minors excluded by C is available.
We believe that these applications demonstrate a lot of potential for elimination dis-tance, and that there are likely many more applications for it.
Algorithmic meta-theorems are algorithmic results that apply not only to a singleproblem, but to a whole class of problems. Our result in Chapter 7 is an algorithmicmeta-theorem, because it applies to all minor-closed graph classes. In Chapter 8 weestablished another algorithmic meta theorem: that any graph problem that is slicewisenowhere dense and slicewise first-order definable is FPT.
Lastly, in Chapter 9 we discussed two problems parameterized by their natural distanceparameters, and showed several fixed-parameter tractability results.
This dissertation has demonstrated that distance is a useful lens through which to
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view the parameterized complexity of graph problems. It certainly has the potential tohelp discover interesting research questions. By being a guide in finding the line betweentractability and intractability for specific problems, it helps improve our understandingof the complexity theoretic landscape.
10.1 Future research directions
In the following we discuss some threads that the research in this dissertation has openedand that we think are worth pursuing in further research.
10.1.1 Elimination distance
We introduced the new parameter elimination distance to a class C in Chapter 5 andthe notion seems to not only leave more room for the study of Graph Isomorphism,initiated in Chapter 6, but also offer promise for defining tractable parameterizations formany other graph problems. These are directions that bear further investigation.
10.1.1.1 Graph Isomorphism parameterized by elimination distances
An immediate question that arises from our work in Chapter 6 is what happens when weconsider other classes of graphs for which Graph Isomorphism is known to be tractable.
For instance, what can we say about Graph Isomorphism when parameterized byelimination distance to planar graphs? Unfortunately techniques such as those deployedin Chapter 6 are unlikely to work in this case. Our techniques rely on identifying acanonical subgraph which defines an elimination tree into the class where GI is tractable,which would be the class of planar graphs in this case. As an obstacle, in the case ofplanar graphs, consider the class of graphs which are a subdivision of K5 (i.e. which canbe obtained from K5 by replacing each edge by a path). It is easy to see that each of thegraphs in that class is deletion distance 1 away from planarity. However, it is impossibleto identify a canonical vertex to delete: the deletion of any vertex yields a planar graph.New techniques are needed to steer around this obstacle.
10.1.1.2 Elimination distance to bounded degree
Another open question raised by Chapter 5 and Chapter 6 is whether elimination distanceto the class of graphs of maximal degree d is FPT. When d is 0, the elimination distanceis just the tree-depth of a graph, and this case is covered by our result in Chapter 8. Agraph of maximal degree 1 is just a union of disjoint edges and isolated vertices, so if C isthe class of such graphs, then we just have edC(G) = td(G) or edC(G) = td(G) + 1. Forvalues of d > 1, it is not clear whether elimination distance is first-order definable. Notethat for d = 2 we have that the elimination distance is almost the generalised tree-depthof G (see Section 5.4.1) and the question is also open.
One possible strategy is to establish that elimination distance to C is first-order de-finable. If C is a graph class with bounded expansion, then so is the class of graphs withelimination distance k to C (cf. Section 5.3), and we might then apply our result fromChapter 8. Indeed, the question for which classes C elimination distance to C is first-orderdefinable is an interesting question. Even more interesting would be negative results thatshow it is not definable.
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But even if it is not possible to establish that elimination distance to a class C isfirst-order definable, it might still be FPT for another reason. So another, more generalversion of the first question is whether for any nowhere dense and first-order definable C,elimination distance to C is FPT.
10.1.1.3 Elimination distance to edgeless graphs: tree-depth
The special case of elimination distance to the class of edgeless graphs is just the tree-depth of a graph. Tree-depth is the special case where the class we are consideringthe elimination to is the class of edgeless graphs. Although it has been studied underdifferent names in the last twenty years now, it has not received as much attention asother parameters such as tree-width. The particular structure of the decompositions, andthe fact that computing the decompositions is itself fixed-parameter tractable, lends itselfto dynamic programming approaches. (See Section 4 for a more detailed discussion.)
What makes the study of tree-depth particularly interesting is that it lies betweenvertex cover number and tree-width as a parameter. Many problems are known to befixed-parameter tractable parameterized by vertex cover number, but either known to behard parameterized by tree-width or with an unknown complexity status. We discussseveral such problems in Chapter 4, but there are many other problems that could bestudied. We mention some examples in Section 4.6.
10.1.2 Algorithmic meta theorems
For a long time the study of algorithmic meta theorems has focused on evaluating first-order logic formulas on classes of sparse graphs. Evaluation of first-order logic on (non-trivial) graph classes closed under taking subgraphs is well understood now: Grohe etal. [92] showed that first-order evaluation is FPT on nowhere dense classes of graphs, andDawar and Kreutzer [110] proved that it is not FPT on somewhere dense graph classes(unless FPT = AW[∗]).
These results for nowhere dense classes are a great success, but a class that trulycaptures the tractable instances of first-order evaluation would have to include somesomewhere dense graph classes as well and would need to satisfy certain closure properties,such as the closure under taking complements (see below).
This has motivated the study of graph classes that are not closed under taking sub-graphs, e.g. graph classes that are just closed under taking induced subgraphs, or graphclasses that include dense graphs? For example Bova et al. [24, 25] and Gajarsky et al. [77]have recently resolved the question for partially ordered sets (i.e. directed graphs that area reflexive, antisymmetric, and transitive) of bounded width.
10.1.2.1 Graph classes including dense graphs
As mentioned above, the result by Grohe et al. [92] applies only to graph classes closedunder taking subgraphs. A more ambitious goal would be to understand what kind ofstructure allows easy evaluation of first-order logic. Ideally we would be able to identifythe class of graphs that precisely captures the graphs where evaluation is FPT. In fact, aclass of structures that does this, would have to be closed under first-order interpretations.For example, the possibility of negating the atoms would mean the graphs have to be
115
closed under negations. This means that sparsity might not be a natural constraint forfirst-order logic.
So another interesting case that seems closely related to our methods, but is not animmediate consequence is that of classes that are given by first-order interpretations fromnowhere dense classes of graphs. For instance, consider the problem of determining thedeletion distance of a graph to a disjoint union of complete graphs. This problem, knownas the cluster vertex deletion problem is known to be FPT (see [99]). The class of graphsthat are disjoint unions of cliques is first-order definable but certainly not nowhere denseand so the method of Chapter 8 does not directly apply. However, this class is easilyshown to be interpretable in the nowhere dense class of forests of height 1. Can this factbe used to adapt the methods of Chapter 8 to this class?
A further step would be to see if there is a nice way to characterise such classes. Aninteresting way to approach this would be to consider the connection of nowhere denseclasses to stability theory noticed by Adler and Adler [5].
Another line of attack would be to use the idea of shrub-depth, recently introducedby Ganian et al. [80], and consider low shrub-depth colourings defined in an analogousway to low tree-depth colourings, which characterise nowhere dense classes of graphs.Such classes are a potential candidate for being exactly the graph classes where first-order logic is fixed-parameter tractable (as opposed to graph classes closed under takingminors / (induced) subgraphs). We expect this would yield a graph class with interestingalgorithmic properties, that would be a natural generalisation of nowhere dense graphclasses.
10.1.2.2 Order-invariant first-order logic on graphs of bounded degree
Another question worth considering is whether order-invariant first-order logic is fixed-parameter tractable on graphs of bounded degree. Order-invariant first-order logic is avariation of first-order logic that allows the use of an order relation in the formulas, butthe truth value of a formula must not depend on the particular order (i.e. it either holdsfor all orders or for none). It was shown that order-invariant first-order logic is strictlymore expressive than first-order logic [94]. A better understanding of this extension wouldbe helpful in database theory, where the physical existence of the database in memoryimposes an implicit order on the database. The database can be understood as a relationalstructure and the user of the database has no control over the order, so order-invariantqueries are particularly interesting.
Grohe and Schwentick [94] showed that whether a tuple in a structure satisfies a querydefined in order-invariant first-order logic only depends on a small neighbourhood of thetuple. This seems to suggest the possibility that order-invariant first-order logic mightbe fixed-parameter tractable on bounded degree graphs as well. However, the proof isvery different from Gaifmans proof and does not involve the translation into a Booleancombination of local sentences, so a new approach would be needed.
10.1.3 Descriptive complexity
There are also several interesting open questions left regarding the descriptive complexityof graph classes featured in this dissertation. For example, it is an open question whether,on nowhere-dense graph classes, MSO2 allows us to express more properties of graphsthan just MSO. Recall that MSO is all of first-order logic extended with set variables
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and quantification over set variables. MSO2 also allows quantification over sets of edges.Courcelle [41] proved that MSO and MSO2 have the same expressivity on k-degenerategraphs, and it would be interesting to extend this to nowhere-dense graph classes.
10.1.4 Subquadratic FPT algorithms
The growth of data sets processed in industry over the last decade has brought newalgorithmic challenges: even algorithms that run in polynomial time, with a polynomialof moderate degree, have formidable runtimes on huge datasets. Motivated by these‘big data’ developments, the threshold for tractability seems to have shifted: A problemneed not be considered tractable on large datasets just because it has a polynomial-timealgorithm. The line between tractability and intractability for such large input has movedto low degree polynomials; in fact, subquadratic computation is the ‘new P’ and problemsthat are provably at least quadratic are the new ‘NP-hard’ problems. The hardness resultsare usually conditional on stronger complexity theoretic assumptions such as the StrongExponential Time Hypothesis (SETH) introduced by Impagliazzo, Paturi, and Zane [101].
The SETH was first used to prove conditional lower bounds for NP-hard problems (seee.g. [117] for a survey in the context of parameterized complexity). More recently lowerbounds for problems known to be in P have been found, often showing that they requireat least quadratic runtime (see e.g. [1, 3, 21, 26, 27, 143, 157]).
Just as for NP-hard problems, there is an interest to find ways around the hardness forproblems that have been shown to be hard (if the SETH is true). Until recently this hasmostly led to the study of approximation algorithms for such problems. Parameterizedcomplexity theory has only very recently been applied ([2, 74, 86]).
The application of parameterized complexity seems very promising because it can bothprovide additional insights into what makes the problem hard (i.e. at least quadratic),and offer algorithms that work well in some circumstances (whenever the parameter issmall). Giannopoulou et al. [86] have formulated a research program to systematicallystudy polynomial-time fixed-parameter tractable algorithms. They have introduced newcomplexity classes. Given a polynomial-bounded function p(n), they define the classFPT(p(n)), the class P-FPT(p(n)) as the problems that can be solved in time O(kt · p(n))for some constant t, and lastly the class PL-FPT(p(n)), defined as P-FPT(n).
An interesting question in this context is whether it is possible to prove algorith-mic meta theorems, comparable to what we present in Chapter 8, to show that certainproblems are in PL-FPT. Given a fixed first-order formula, it can be evaluated in polyno-mial time on a graph and the problem of evaluating a given first-order formula is fixed-parameter tractable parameterized by the size of the formula on nowhere dense classes ofgraphs. Is there maybe a fragment of first-order logic that can be evaluated in linear FPTtime with only polynomial parameter dependence?
What about the evaluation of a fixed first-order formula, and distance parametersto sparse graph classes? Could it be that the first-order model checking problem of afixed formula, parameterized by the tree-width of the graph, is in PL-FPT? Or, muchsimpler, parameterized by the vertex cover number? Unfortunately this is not the case ifthe SETH holds, even for first-order formulas with just 3 variables. To see this, considera graph construction from another hardness result: Abboud et al. [2] argue that there isno subquadratic fixed-parameter algorithm with subexponential parameter dependencefor the Diameter problem unless the SETH is true. They use a reduction from CNF-
117
SAT to Diameter, discovered by Roditty and Vassilevska Williams [143]. A look atthe reduction reveals that given a CNF-SAT instance with n variables and m clauses weobtain a graph with O(2n/2+m+m) vertices and O(2n/2 ·m) edges.
Abboud et al. [2] show that if there is an algorithm that solves the Diameter problemon a graph with b edges and some parameter k in time O(p(k) · b2−ε), then CNF-SATcan be solved in time O(p(k) · poly(m,n) · 2n− εn2 ), which implies that the SETH is false ifp(k) is subexponential in m and n.
A closer look at the reduction in [143] also reveals that the constructed graph has tree-width m and in fact vertex cover number m. This implies that the Diameter problem,parameterized by tree-width or vertex cover number, is not in PL-FPT unless the SETHis false.
Moreover, the reduction does not require us to solve the whole Diameter problem.It is sufficient to distinguish the cases where the diameter is 2 or 3. Of course ‘diameter6= 2’ can be easily expressed in first-order logic; all we need to say is that there is a pairof vertices that are not equal, not adjacent, and also not connected via a third vertex, orformally:
∃x∃y.x 6= y ∧ ¬E(x, y) ∧ ¬ (∃z.E(x, z) ∧ E(z, y)) .
This implies that model checking a fixed formula from 3-variable first-order logic pa-rameterized by tree-width or vertex cover number is not in PL-FPT unless the SETH isfalse.
Proposition 10.1.1. Evaluating a fixed 3-variable first-order formula on graphs is notfixed-parameter tractable parameterized by the vertex cover number of the graph, unlessthe SETH is false.
However, the research of the parameterized complexity of problems in P has just begun,and there are many interesting questions to study, including potential meta theorems.
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Index
Ck, 63EG(v, S), 27M(C), 29Nr, 27[·]fpt, 25#G, 82FPT, 24Forb(M), 29Πi, 23Σi, 23W[i], 25XP, 25χp(G), 31χv, 82edd(G), 65`-clique dominating set, 106`-clique domination number, 106level<(v), 32∇, 29, 30`dens(C∇), 30pw(G), 36v, 82vF , 28tw(G), 36vc(G), 35b-anchored k-core, 109b-box representation, 49d-degree torso, 79d-deletion set, 74k-constructible, 37k-core, 39k-core number, 39k-degenerate, 39r-neighbourhood, 27|G|, 26||G||, 26Anchored k-core, 109Boxicity, 50
Clique Dominating Set, 106Compute Quasi-Wide Parameters, 107Deletion Distance to Q, 98Distance Independent Set, 96Elimination Distance to Excluded Mi-
nors, 89Graph Isomorphism, 71List Colouring, 46Multicolour Clique, 47Pre-Colouring Extension, 46RCE(S, r), 101Regularity Constrained Editing(S, r),
101WDCE(S), 100Weighted Degree Constrained
Editing(S), 100Weighted Fagin Definability
Problem, 25
abbreviations for first-order logic, 22additive p-approximation algorithm, 49anchor set, 109anchored k-core, 109apex, 62apex graph, 62apex graph class, 62apex vertex, 62arity, 21atomic
formula, 22
bag, 36bound, 23bounded expansion, 30bounding set, 75boxicity, 49
canonical form, 28chain, 31
131
class of sparse graphs, 33clique dominating set, 106clique domination number, 106clique-width, 37closure under disjoint unions, 62colouring, 28component, 27connected component, 27connection closure, 91core, 39covering relation, 32covers, 32
degeneracy, 39degenerate, 39degree, 27deletion distance, 34deletion distance to a class C, 34deletion distance to degree, 74depth-r minor, 29diameter, 27directed graph, 26disjoint union of graphs, 62distance parameter, 33dominating set, 106domination number, 106
eccentribity, 27edge-degree, 101edge-degree-regular, 101edge-regular, 101edit distance, 35edit distance to a class C, 35effectively nowhere dense, 29elimination distance, 57, 89elimination distance to bounded degree, 65elimination distance to degree, 65elimination order to a class, 58elimination order to degree, 66elimination-order, 32expansion, 22
first-order formula, 22first-order formula with free relation vari-
ables, 23fixed-parameter tractable, 24
model theoretic, 26formula, 22free variable, 23
generalised tree-depth, 45graph, 26
height of an order, 32
independence sentence, 95independent set, 27induced subgraph, 27induced substructure, 22intersection of graphs, 49interval graph, 49isomorphic coloured graphs, 28isomorphic graphs, 28
kernel, 24kernelization, 24
language, 24level, 32, 50linear order, 31low tree-depth colouring, 31
max leaf number, 38minor, 29minor map, 29minor-closed, 29monadic second-order logic, 23MSO, 23
neighbourhood, 26neighbourhood cover, 95non-maximal subgraph, 61nowhere dense, 29number of connected components, 82
parameter function, 97parameterization, 24
model theoretic, 26parameterized approximation algorithm, 49parameterized problem, 24partial order, 31path, 27path decomposition, 36path through a subgraph, 27path-width, 36pre-order, 31problem, 24
model theoretic, 26proper colouring, 28pruned tree-depth decomposition, 50
132
quantifier rank, 23quasiorder, 31
radius, 27reachable, 27reduct, 22relativisation, 98
scattered, 30second-order formula, 23second-order logic, 23sentence, 23set of minimal excluded minors, 29shallow minor, 29signature, 21slicewise first-order definable, 97slicewise nowhere dense, 97solution, 23solution of a formula, 23somewhere dense, 30spanning tree, 38sparse graph class, 33sparsity, 33Splitter game, 96strongly-regular, 101structure, 21subdivide, 38subdivision, 38subformula, 22subgraph, 27
term, 22theory, 23torso, 79total order, 31tree decomposition, 36tree-depth, 34, 41, 43, 102tree-depth colouring, 31tree-depth decomposition, 42tree-order, 32tree-width, 36type of a node in a tree-depth decomposi-
tion, 50
uniformly quasi-wide, 30universe, 21
vertex cover, 35vertex cover number, 35
vocabulary, 21
WDϕ, 25
133